I don't know about the A and the E, but upside down V is the very real capital Lambda from the greek Alphabet while the upside down L is the capital Gamma from the greek Alphabet. We just like greek symbols for the most part :p
The way my teacher explained the implication and its truth table is as follows. Suppose I say "If I win the lottery, I will buy you a house!" Logically, this is saying P => Q where P is "I win the lottery" and Q is "I buy you a house". Now think about the following: in which cases are you satisfied? When P is true and Q is true, then I kept my promise and you're happy. So T => T is T When P is true but Q is false, then I broke my promise. I won the lottery, but didn't buy you a house. You're angry, sad, dissapointed. So T => F is F When P is false, I haven't really made any promises. I never said what I'd do if I did NOT win the lottery. So, if I still buy you a house, you're definitely going to be happy, but even if I don't, you won't be mad because I didn't win the lottery. Hence, F=>T and F=>F are both T.
That's similar to how I explain it to my students. Suppose I claim "If you do your homework, I'll give you an A". In which situation could you claim I lied? Only in the situation where you did your homework but I didn't give you an A.
I still think my "tennis on a cylinder" idea fits better.. only one way fails to cross an imaginary line, the one where the two guys are actually playing back and forth normally, instead of around the world (which in all other cases will cross the line) And T throws rights and F throws left. F left, because it looks more like an L with a beard.
When I was learning this, the hard part for me to digest was that implication is a logical operator. For a while I was thinking about it like "this symbol is used as equals AND an operation?!". Nowadays, whenever I try using any complex logic with my friends I always say something like "We'll assume it's true, because it doesn't matter if it isn't", because it feels like that mindset is what "non-math" people struggle with, and many "math" people take for granted
I think the hard part is that thing we don't typically think of as true-or-false statements are so in logic/math, because everything is. Like if you write x=3, you are actually saying "the statement x = 3 is true", except the whole "is true" part is implicit. However, this is fine because setting x to a value is an intuitive enough concept you can just think of it that way. It gets weirder though with things like implications, where we especially don't think of those as true or false. Without a background in math, I feel like a lot of people would be confused if you asked "if x then y, is this true"? Once you get a grasp on the implicit things, it starts to make more sense
That "Is it a boy or a girl? Yes" joke reminded me how in the national math team training camps we do this joke all the time, some asks for example "Wait so is lunch now or do we have a lecture?" and someone else responds "Yes", man that does not get old
Having studied Maths at uni, I saw this thumbnail and thought 'YEAH?? OBVIOUSLY??' Then I actually watched the video and it's a really good video explaining the basics. Nice!
I loved the joke about "THERE EXISTS" as someone did the same for the factorial in my class some days ago 😂 More seriously, I really enjoy your videos, they're very recognizable because of their graphic identity and the music behind, and your way to show examples to be very clear, to stick little good-looking papers and to write on a black board, it's very pleasant! I particularly loved your series about foundations of numbers, but this video about logic was very good as well and I appreciated it! Continue like that!
Having never studied this kind of math yet having years of experience in the field of programming, it's incredibly interesting seeing how a lot of concepts in both fields are equatable.
@@mbdxgdb2 bro he just said " a lot of concepts in both fields are equatable.". Which mean he might have studied it in programming but not through math.
I'm about 6 minutes into this video and I can't unsee the similarities(atleast so far) with logical math, and programming operators. I think I might be able to understand this.
For every upside down A there exists a backwards E ;) In my career as a software engineer you of course lived and breathed this. You may not express it formally but logic was your constant companion.
Be careful when translating natural languages into logic: people often can be tricky. "Everybody loves somebody" seems to have an obvious meaning. But it could either mean "There exists one person that every person loves" or "Every person has (at least) one person that they love." (I find it fun to deliberately misinterpret ambiguous sentences.)
That reminds me of the Beatles song "All you need is love". People tend to hear that philosophically as something like "the only thing that any person actually needs in life is to be loved", whereas I believe the song was actually intended to mock consumerism and greed and actually meant something closer to "you already own literally everything material and/or of financial value, and now the only thing you are still lacking is love".
@@Repsack2 The Chuckle Brothers used to end their live shows by asking people to drive carefully, saying... Paul: On your way home please take care, as statistics show that a man gets knocked down every other night of the season Barry: Yeah, and he's getting really fed up of it now!
I'll just add this because it's something that really clarified the existential and universal quantifiers for me: the existential quantifier creates a giant OR statement, and the universal quantifier creates a giant AND statement. For example, let the universe of discourse be {0,1,2,3,4,5}. Then: For all x: (x > 3) 0>3 and 1>3 and 2>3 and 3>3 and 4>3 and 5>3 false. There exists x: (x > 3) 0>3 or 1>3 or 2>3 or 3>3 or 4>3 or 5>3 true.
@@quantumgaming9180 I think it was at least a year or multiple years between the time I was introduced to the quantifiers and the time I found out they were equivalent to AND statements or OR statements.
In fact there are alternative symbols for quantifiers: ⋀ and ⋁. In the same way ∏ and Σ mean product and sum over elements in a set, ⋀ and ⋁ mean conjunction and disjunction over all elements in the set.
Great video! I seriously went from seeing an incomprehensible mess to thinking “well yeah, obviously”. I’m a fan of math, but never felt I had enough talent to go get an advanced degree in it. But your videos make these esoteric sounding ideas easy to grasp. I would love it if you covered Gödel’s incompleteness theorem at some point. Love your work!
Wow I finally get the implication part. Looking at 11:06, If P is a circle inside Q in this 'space of all possibilities' then you can point you finger at any point on that space and say: Point x is inside P and Q Point x is outside of P but inside of Q Point x is outside of P and outside of Q But you cant point to a space that is inside of P and outside of Q. Thats why that is F, its an impossible state. In the case of "greg is a cat->greg is a mammal" the only contradiction is where greg is a cat and not a mammal. No other scenario is contradictictory.
I think your reciprocal proposition is actually a really strong argument for why implication is the way that it is. For all real numbers x, (if x is not 0, then there exists a real number y such that xy=1) We really want the implication to be true for all x for our universal quantifier, but there is a value of x where the first statement of the implication is false. The thing we're trying to prove doesn't really care what happens when x=0, but we still need the implication as a whole to be true for all x, including 0, the thing we were trying to exclude. So, we just define that case to be true no matter what, because it means we don't have to worry about it breaking our quantifier.
Never understood those weird math symbols but this video really helped. Also as a Software Developer I can find many similarites in the language of maths and code.
Very good video! I think that the solution to the exercise at the end of the video is this 1: true (ironically this is the only one i'm unsure about) 2: false (because of 0) 3: false (because of 0) 4: true (because there's 2) 5: false (there isn't a value that works for every y) 6: true (for every x there is that works) 7: true (i think this doesn't need explanation) 8: false (because there are numbers that aren't elements of Q and their sum is an element of Q: π and -π if you sum them you get 0 which is an element of Q 9: true (because R is dense) 10: true (the only value is 0)
@@ziadhossamelden9241 there isn’t a value that added to any y equals 0 The proportion says that a value that works with any of the real numbers but there isn’t because for examples for 3 only -3 works such as 3+(-3)=0 but it doesn’t work for -4
I was looking for this comment, I have a question, tho: 1: Does this mean we all think that 0 is even? It follows the pattern, but it's just kind of weird lol. And I got 8 and 10 wrong. Both were obviously you're answer after thinking about them further. I didn't think about transcendentals for 8, and I didn't think about how the uniqueness of 0 is the special characteristic that makes statement 10 true.
@@kindlinEvery even number "x" is a multiple of 2, which means you can write it as x=2k where k is an integer. 0 is obviously an integer and 2*0=0 => 0 is an even number.
This video is delightful. Mathematics would probably be more palatable to a general audience, if children were given a helpful introduction to logic in the early days. Most of the benefit of mathematics in real life is logic. Many people who struggle with math either fail to see its relevance or fail to grasp the basic logic underpinning the statements.
1. True. All integers are either odd or even. This is a direct consequence of the Theorem of Euclidean Division, which states: For every pair of integers m,n, there exists a unique pair of integers q,r, with r < n, such that m = qn + r. In this case, n = 2; therefore, all integers can be expressed as either 2n or 2n + 1. 2. False. There exists a real number, namely 0, such that it is neither positive nor negative. This follows axiomatically from the fact that the set of real numbers is an ordered field. 3. False. There exists a real number, namely 0, such that 0 is not positive, but 0 is not negative. This is equivalent to the previous proposition. 4. True. There exists a natural number, namely 2, such that 2 is prime and also even. 2 is prime beacuse it cannot be expressed as a product of two smaller natural numbers. Being the first natural number greater than 1, the only possible "product of two smaller natural numbers" is 1x1 = 1, not 2. 2 is even because it can be expressed as 2 = 2x1, that is, 2 = 2n for n=1. 5. False. There exists NO real number which has the property of "destroying all numbers" through addition, that is, that the result of it added to any number will always result in 0. To prove this, suppose, by contradiction, that such a number x exists. That is, x + y = 0, for any real number y. Then, take y + 1: x + (y + 1) = (x + y) + 1 = 0 + 1 = 1 =/= 0, which is a contradiction. (On the other hand, there exists such a number for multiplication: 0 "destroys all numbers" through multiplication since y.0 = 0, for any real y.) 6. True. For every real number x, there exists a real number y, called the "additive inverse" of x, with the property that x + y = 0. This number is y = -x. This is a property that defines the set of the real numbers as a field. 7. True. To prove this, consider the second member of the "and" relation: x² = y². By subtracting y², we have x² - y² = 0. Factoring, we have (x - y)(x + y) = 0. A product of two real numbers is zero if, and only if, one of the numbers is zero. Therefore, either x - y = 0, which would mean x = y (not allowed by our premise), or x + y = 0, which would mean x = -y. Therefore, the implication holds. 8. False. P: x is rational and y is rational. Q: (x+y) is rational. Q does not imply P: this means that, if (x+y) is rational, then x and y need not be both rational. In fact, for x = √2 and y = -√2, we have (x+y) = √2 - √2 = 0 rational, but neither x nor y is rational. 9. True. In fact, we can take z = (x+y)/2 which has the required property: z - x = (x+y)/2 - x = (y-x)/2 which is positive when x < y, meaning x < z. y - z = y - (x+y)/2 = (y-x)/2 which is positive when x < y, meaning z < y. 10. True. That unique number is 0. It is true that 0 has the required property, since for y > 0, 0² = 0 < y. The proof that 0 is the unique number with this property is as follows: Suppose, by contradiction, that another number x =/= 0 has the same property. Then, x² > 0, which implies x²/2 > 0. Take y = x²/2. y > 0 but x² > y = x²/2, which is a contradiction. (This proof requires the knowledge of the fact that: x² = 0 iff x = 0, x² > 0 otherwise.)
Beautiful list! I also have a question about 7, but for a different reason. I believe that for all real numbers x,y, if x = -y, then x ≠ y and x² = y² is also true, meaning it has a two way relationship. Is it a problem if one way implication is stated for a two way relationship like that? After typing it, my gut is saying that (a b) ==> (a ==> b) (I hope I did that right 😅), but I am unsure. Do you know the answer to this? I also need to ask if 0 counts as an even number for the response to question 1. I believe it is a counterexample to the statement.
I was always curious about logic notation. Now I won't have to be haunted with pages and pages of unknown symbols when I choose to study this subject. Very good video!
22:28 another way to prove that one false would be, since x is an element of the real numbers, and y covers all the real numbers, y also covers x, and x is never less than itself
26:37 1. true (either have to be true) 2. true (same as last) 3. true (not positives are negative, but the sign could also be ) 4. false (no prime is even since even is divisible by 2) 5. false (**BOTH** **MUST** be 0, and Ay e R is not only 0) 6. false (same as last) 7. true (the only number squared that isnt itself is its negative) 8. true (both x and y must be in Q for the x+y to be in Q) (EDIT: false, irrational+irrational can equal a rational, e.g. π+(1-π)=1) 9. true (you can always fit a real number between 2 other) 10. false (no one REAL number squared is < 0)
At first,i used to find logic math/logic philosophy hard because i thought that it was for prodigies or geniuses,well basically,i'm good at math,but i was not that good at logic math,i only knew the element of,and the sets,but because of you, I'm starting to love logic math,and it got easier for me,and I'm starting to get hooked up with it,thank you😊.
A major factor in the confusion of the or statement is the implied use in natural language of "or" as "exclusively or". With the drinks example, I don't know if I'd be happy being served two hot drinks. Those kinda have a time limit.
I'm a tenth grader and I understand the notation. Before I actually watch the video, here is how I would say that example statement: "For every real x that is nonzero, there exists a real y such that xy=1" I thank my math teacher for teaching at above grade level, and we also strangely learned formal logic and its notation in philosophy classes 👍
Wait did the joke at 6:21 just go over my head, or does UK English keyboard layout actually have the logical negation symbol on it? I get the feeling I missed the joke but it would be cool if that symbol was actually standard on a keyboard
I didn't realise this was only a UK thing! But yeah our key below Esc and to the left of 1 as the ¬ symbol on it. As well as ` which only TeX-savvy people use. And also ¦ which... your guess is as good as mine.
@@AnotherRoof Oh interesting! Yeah, on an US English keyboard layout, that key has ` (backtick) also, but the other symbol on it, which you reach with shift, is ~ (tilde). But you say there's a third symbol there? I recognize that broken-vertical-bar but the US layout only has the regular vertical bar | on the same key as the backslash (under backspace)
@@Scum42 Yeah I realise that now -- we also have the vertical bar on the backslash key but it's left of Z. I do remember some of the differences now because I once bought a US-layout keyboard but I like our big, tall 'enter' key, But I never realised ¬ was a UK thing. Don't know why -- literally no idea what it's used for outside of formal logic!
Lovely video! My exposure to logic has been in computer programming, really neat to see the parallels!! edit: my logic answers 1) If x is in the set of Integers Z, the Odd set holds x or the Even set holds x. This is True (assuming 0 has parity) 2) If x is in the set of Real Numbers R, the Positive set holds x or the Negative set holds x. False (assuming 0 is Real and Unsigned) 3) For every value x contained in the set of Real Numbers R, if Positive doesn't hold x then Negative does hold x False (assuming 0 is Real and Unsigned) 4) There exists some value x in the set of Natural Numbers N where x is prime or x is even True (This will work for any prime or even number) 5) There exists some value x in the set of Real Numbers R, which for every value y in the set of Real Numbers x+y=0 False (by contradiction: x:4+ y:3 ≠ 0) 6) For every value x contained in the set of Real Numbers R, there exists some value y contained by R in which x+y=0 True (Any number's opposite added to the same number will yield 0) 7) For any two values x,y contained by the set of Real Numbers R, if x ≠ y AND the square of x is the square of y, that x = -y True (if x and y are not equal but their squares are the same, then the magnitude of x and y must be identical. X^2 = Y^2, X = ±Y) 8) For any two values x,y contained by the set of Real Numbers R, that if x and y are both contained in the set of Determinate Fractions Q, the sum of x and y must also be contained in the set of Determinate Fractions Q (and vice versa) True (the sum of two fractional numbers will never yield a non-determinate fractional number, and the addends of a fractional number will always be two determinate fractional numbers--otherwise the definition of a detemrinate fractional number breaks) 9) For any two values x,y contained by the set of Real Numbers R, if x < y, then there exists some number z contained by the set of Real Numbers R that falls between x and y. True (Real Numbers allows non-wholes, and a non-whole number can __always__ be subtracted or added to. An easy way to guarantee this is by picking the minimum place value of x and y together, making it one order of magnitude smaller, and adding a single unit of that place value to x) 10) There exists a unique value x contained in the set of Real Numbers R, that with any value y contained in the set of Real Numbers R, wherein if y > 0, the square of x will be less than y False (by contradiction: More than one value. x:2^2 < y:5 and x:2^2 < y:6, therefore x:2 is not unique) Super super fun brain teasers!!!!! My favorite was number 8 and I do hope I'm correct on these. Thanks for a fantastic video. edit edit: somebody added an irrational number [(π) that can be a determinate fraction] to itself on #8 and proved me wrong. Cheers!!!
For #4, you used the wrong connective; properly it should be "There exists a natural number x such that x is prime and x is even." Which is true, x=2. For #5, you're correct, but you your proof is not a proof by contradiction, and isn't sufficient to prove the statement true or false, because a claim is being made about a property of all real numbers. A proper proof by contradiction here would be something like y = 1-x, x + (1-x) = 1, 1!= 0. For #8, even considering your edit, the rationale is wrong. Q is the set of rational numbers; π is not a rational number, nor is π + π, so plugging it in for x or y creates a vacuous statement and doesn't prove anything. A better example would the counterexample x=π, y = 1-π. x+y=1, which is in Q, but neither (x in Q and y in Q) does not hold, so the biconditional is not satisfied and the statement is false. For #9, you're correct, but I think a better explanation is that it's possible to define z such that the value of z always falls between x and y; the simplest example I can think of is z = (x+y)/2. For #10, you're misinterpreting the meaning of ∀. "∀y ∈ R" means that the proposition must be true for all values of y, not for any single value of y. The statement is true; x=0 is the unique value whose square is smaller than any positive number. I think the best way to think of ∀ and ∃ is that in both cases, you must consider every possible value of the variable. For ∀x, the predicate must be true for every possible value of x, but for ∃y, you only have to prove that out of every single possible value of y, the predicate is true for at least one. I would try to avoid using "any" in phrases like "for any value" because that usage is ambiguous; "for any value" could mean that we should be able to plug any conceivable value in and the statement is true, or it could mean that we want it to be true given at of the set. For example "x+1=2" is true for "any" real number because it's true for 1, but it's not true for "any" real number because it's not true for 2.
Also for #8, another way to prove it false is that if you let (x ∧ y) = 1/2, then x + y = 1 which is not within the set of rational numbers. Likely, if x + y ∈ Q, then it doesn’t mean that (x ∧ y) ∈ Q because you can let x = 1 and y = 1/2 which means x + y = 3/2, and even tho the sum is rational, its components x and y are not. 🙏
Using mathematical symbolism and logic can provide a powerful bridge to connect theological/metaphysical concepts with scientific/physical descriptions in a rigorous way. Instead of relying solely on binary true/false valuations, engaging non-contradictory/contradictory modes of reasoning could be highly fruitful. Here are some thoughts on how we could apply this approach: 1. Multi-valued and Fuzzy Logics Rather than classical bivalent logic, we could explore multi-valued algebraic logic systems that allow for more nuanced truth valuations beyond just 0 and 1. This could capture theological notions of paradox, ineffability, and transcendent reality that goes beyond strict binarization. Fuzzy logics which admit truth values in the continuous range [0,1] could model metaphysical concepts that are irreducibly vague or context-dependent. Non-contradictory/contradictory could then be represented by sub-ranges of the multi-valued domain. 2. Paraconsistent Logics Paraconsistent logical systems are designed to deal with contradictions in a controlled, discriminating way rather than just admitting logical explosion. This could allow rigorously reasoning about metaphysical statements that are paradoxical or logically inconsistent from a classical perspective. Non-explosive paraconsistent frameworks like relevance logic could formalize theological ideas involving prescribed inconsistencies or contradictories without trivializing the entire system. Non-contradictory and contradictory conditions could be encoded precisely. 3. Modal Logics and Intuitionistic Systems Modal logics explicitly capture notions of necessity, possibility, and ontological modalities. We could use graded/fuzzy modal systems to represent transcendent, ineffable realities beyond typical ontological constraints. Intuitionistic logics based on constructive reasoning avoid strict bivalence and the principle of excluded middle. This could model metaphysical concepts that are not straightforwardly decidable in a binary fashion. 4. Substructural Logics and Resource Semantics Substructural logics like linear logic impose resource-consciousness by controlling structural rules like weakening and contraction. This limited, pay-as-you-go approach could capture theological ideas of existential scarcity, ontological austerity, and irreducible indeterminacies. Phase semantics and resource models in these logics could provide novel metaphysical interpretations and construct ontological stances beyond strictly bivalent modes. 5. Topological Semantics and Cohesion Cohesive topological models using homotopy theory and algebraic topological semantics could provide a powerful geometric metaphor for non-contradictory/contradictory conditions in terms of intrinsic continuities, boundaries, and points of inflection. This could unify metaphysical and scientific descriptions by embedding them in a common topological setting where contradictions are smoothly navigable via continuous pathways rather than pure bivalence. By leveraging the immense richness of mathematical logic and non-classical reasoning frameworks, we could indeed use symbolic representations to bridge theological abstractions and physical observations in a philosophically robust yet scientifically grounded manner. The non-contradictory/contradictory mode could become a new conceptual lens, expanding rigid true/false binaries into a continuum of coherence where metaphysics and science fluidly intersect. I'm happy to further explore concrete examples of how to apply these ideas to specific theological/metaphysical notions and their scientific counterparts.
In language, we tend to use "or" to mean "xor" or "exclusive or". This version is true when one of the inputs is true, but not both. This is why the "yes" to "or" questions play as jokes.
@@Anonymous-df8it No, I would have to say, “either x or y, but not both,” not just “either x or y.” However, we often do say “or” when we mean “xor.” Do you want steak or chicken? Yes, both, please.
@@bethhentges a) Why wouldn't "either x or y" be sufficient? b) "However, we often do say “or” when we mean “xor.”" Your example question isn't meant to be taken literally; even if you interpret the or as xor, you still don't get the intended meaning (see 'is it a boy or a girl?')
Weirdly enough, a few days ago, I was thinking in bed "How would I introduce the concepts of Boolean logic to a middle school/high school class?" (I am totally serious) Your video tracked almost precisely with the way I would have laid it out (I didn't go into quantifiers, but I did cover a few things like DeMorgan's Laws, and various alternate notations). otoh, you taught me something I did not know (or at least remember?), the uniqueness quantifier ∃!
Turning the lights on by flipping the wall switch, and then turning them off. 1+1=0 Compare two circuits: one in series, one in parallel. Then put a switch in each circuit.
I know us French people tend to make everything different when it comes to maths, but I just wanted to tell you that for us, 0 is always a natural number and 1 is never prime, except if you wanna define it otherwise but I've never seen anyone do so yet
For some reason in the USA, in K-12 ed, and the first two years of college, we make a distinction between natural numbers (positive integers) and whole numbers (non-negative integers). Then once you are in your third yr at college and start group theory/abstract algebra, then we change the definition of natural number to include zero. In the USA, the number written -3 is “negative three,” NOT “minus three.” The word “minus” should be used only for the operation of subtraction. In everyday life, we often hear “minus” used incorrectly as “negative.” Also, in the USA -3 is an integer, but it’s not a whole number, because the whole numbers are the non-negative integers only. I tell my students that definitions develop over time. They start as a general description, and they get more precise as the object becomes more understood. Along the way, “edge cases” are sometimes included and other times not. It’s important to know what those edge cases are so that when you engage with a new person/course/text, you will know you need to agree as to whether or not the definition is inclusive of the edge case or not. For the purpose of the new discussion we need to know: Is zero a natural number? Can a line be parallel to itself? Is a rectangle a trapezoid? When we say suppose a and b are two _____ , are we allowing them to be the same _____ , or are we assuming they are distinct? Regardless of which choice we make, we need to keep that in mind as we go forward in the statements of new theorems and definitions.
So glad to hear I'm not the only one who remembers the AND symbol as the n in fish n' chips (it's also how i remember the difference between union and intersection in set theory)
For the cover: (btw I’m 12 but still understands this due to knowing Python and JavaScript) If any number (variable x) that is real number and not zero, there will be at least one real number (variable y) that when multiplied to x, result becomes 1. Example: x = 5; y = 0.2; xy = 1 x = 0.01; y = 100; xy = 1 x = -255; y = -255; xy = 1 x = pi; y = 1/pi; xy = 1 x = 1.23e+300; y = 1.23e-300; xy = 1 x = cos(0); y = cos(0); xy = 1 x = 69420; y = 1/69420; xy = 1 Invalid because they’re not real numbers or is 0 x = infinity x = i x = 0
27:03 funny how "and/or" was part of the ending. I guess outside mathematical logic "or" is more commonly understood as "xor". If we apply that to the frase it could be "and + xor", or just "or". From a boolean point if view, I think "or" is enough, but to comply with common practice for language and communication it is better to use "and/or"
@@Anonymous-df8it yes 'exclusive or' is fairly easy to express in words. "Ether A or B" is very specific (aka. "xor" in boolean algebra). Similarly apply for and. "A and B" is very specific. (aka. "and" in boolean algebra). My primary focus was possibly was lost (in cyberspace), because of other matters matters mentioned in the short message; and it's difficult to convey what's emphasized as primary focus in text. I can't come up with a short and umabigious way to say 'inclusive or' in words; to prevent misunderstandings. "A or B or both" is on way, but I think it's longer than I want. "A and/or B" is short, but feels 'messy'.
@@Anonymous-df8it The slash is often read as "or" when used with other words, but I think saying "A and-or-or B" is kind of a messy and kind or confusing tongue twister.. Of course you can say "A and-or B", and that's possibly the way most people would say it. 'messy' as in cluttered with different things, because it's difficult to use only words (plain lerrets), but easier to convey by mixing in special characters
This is the part of maths that actually feels like learning a language, and being able to just translate it into an English sentence is very satisfying
haven't finished the video yet but im trying to apply what i know so far by trying to define the XOR operation: R XOR Q = (R∨Q)∧￢(R∧Q) reasoning: in XOR, one of R and Q has to be true (the first term) and they cannot be both true (the second)
I dont have this keys on the keyboard so I use ! for not, & for and, | for or as you'l see in programming the first way is the formal along with (!R&Q) | (R&!Q) the second way is just a reflaction of the fact that "if and only if" means they are the same which is what the xnor gate checks, and xor is not xnor but xor also have a diffrent symbol which is a + in a circle
OMG you're so helpful your video is really interesting and informative. Also love the jokes ,it break the tense that built with in me every time the problem and the materials getting difficult. Ty kind sir. You're Video is Amazing all love from me you sure put a lot of care into it♥🥰.
The way I would explain implication is basically that, because of the law of the excluded middle, P implies Q still has to have a truth value when P is false. Either way could work, but saying implication is true when P is false has less weird implications. It really should be "indeterminate", but that's not an option.
For all the programmers watching this video marvelling at the similarity with various programming languages, that is no accident. Programming languages were basically conceived as a way to implement first order logic.
0:40, well it might not have been geared for me, but somehow I had never come across that use of “!” to mean “a unique” before, so I learned something! :)
omg thanks for explaining this in such an understandable way... a lot of these ideas I already kinda knew from functional programming concepts (the "for all" and "there exists" seemed very familiar like the .all() and .any() methods for iterators in Rust) but I had no clue how people describe them in math terms. Especially that "implies" part was sorta tricky, but that circle diagram was pretty helpful.
I have almost finished my CS degree and never understood the implication and this guy managed to explain it to me, what a guy, implication is one of the most important "tools" in math an it made me always feel insecure because i've never understood it correctly, thank you very much
'so long and/or farewell' since 'and v or' has the same truth table as or, and equivalent statement should be 'so long or farewell' ps not sure if my language is mathematically rigorous or not. if error lemme know tq tq
Can you do a video explaining all 16 diagrams of VENN-diagrams, in relation to Binary numbers and logical reasoning. So, I would like to entirely understand why the Venn diagrams are the best way to use for logical reasoning. Thank you very much in advance.
This video is very nice for beginners, I'll probably recommend this video to students at the beginning of a Math proving class so they know what they'll be getting into.
24:08 wouldn't the opposite of the inside statement also be true? if there's a real number y such that xy = 1, then x must not be 0. Wouldn't that be an "if, and only if" statement then?
I'm 33 seconds in but i remember my formal logic course from first semester uni. This says: "For all x which are an element of the real numbers, if x is not equal to zero, then it follows that there exists a y which is element of the real numbers, such that x times y is equal to 1." The upside down A fittingly stands for "for _all"_ and the mirrored E stands for "there _exists"_ .
For those who want an easy translation, the intro statement says “for all x that exist in the real numbers, so long as x does not equal zero, then there exists a number y which when multiplied to create xy, you get 1.”
My attempt at the 10 questions: 1. True, all integers are either odd or even. 2. False, 0 is a rational that is neither positive or negative. 3. False, again 0 is not positive and also not negative. 4. True, 2 is prime and even. 5. False, there is no number that is 0 when anything is added to it. 6. True, y=-x 7. True, I can't think of any counterexamples. 8. False, there can exists two non-rational numbers that add to a rational number, (pi)+(1-pi)=1 9. True, you can take the average of x and y to get z. 10. True, x=0.
Nice job! 👍 I'd just like to add the following note: Just because you are unaware of any counterexamples doesn't immediately imply that the statement must be true. Though, luckily for you, #7 can be proven to be True because if x² = y² then x² - y² = 0. Applying the difference of squares and solving for the cases where x ≠ y gives us the desired x = -y. Hope this helps, and stay curious! :)
As the relatively fames phrase goes "One man's modus ponens is another man's modus tollens, another man's disjunctive syllogism or another man's indirect proof.". Since these four logical expressions are logically equavalent to each other: [A⇒B] ≡ [¬B⇒¬A] ≡ [¬A∨B] ≡ ¬[A∧¬B] the corresponding syllogisms are then also logically equivalent to each other: (P1) A⇒B (P2) A (C1) B [from P1 and P2 by modus ponens] (P3) ¬B⇒¬A [from P1 by contraposition] (C2) B [from P2 and P3 by modus ponens] (P4) ¬A∨B [from P1 by material implication] (C3) B [from P2 and P4 by disjunctive syllogism] (P5) ¬[A∧¬B] [from P4 by de Morgan's law] (P6.0) ¬B [indirect proof assumption] (P6.1) A∧¬B [from P2 and P6.0 by conjunction introduction] (P6.2) [A∧¬B]∧¬[A∧¬B] [from P5 and P6.1 by conjunction introduction] (C4) B [from P2, P5 and P6.0-P6.2 by indirect proof] So I guess, that these are all from the same partition of syllogisms, which we might call the simplest valid derivation from two premises. Just Another Roof for you.😉
I keep thinking that implication is just an artifact of the premise that truth conditions can only be either True or False. Functionally, A -> B is equivalent to ¬A V B, which is easier to interpret. Though the arrow makes for a nice shorthand. Considering the alternatives, saying an implication is false when the condition in not met just gives you conjunction. Any other alternative gives you the identities of the inputs. But realistically, I think what people would _want_ to say is that an implication whose condition isn't met is Unknown "U", rather than True "T". That's just not an option, though. Say one wanted to represent A -> B with sets. They could draw A as a set subsumed by B, but that carries the subtle additional idea that A exists. It's the same discrepancy with "The king of France is bald." Does that mean that the king of France _exists_ and is bald, or only the implicational relationship? Similarly, What is the probability of B given A, if the probability of A is 0? Freaky stuff...
Since implications are themselves either true or false, does that mean we can nest them? For example ( P => Q ) => (¬Q => ¬P ) which should be true for every proposition P and Q
The end problems, 5 is true but 6 is false right? Also what is the distinction between just stating x belongs to R and using for all x belongs to R? In logical statements, the're treated the same right? Getting this inquiry from the 8th question in the end slate, it seems redundant to have specified R when you then go on to specify Q
Check your answers for 5 and 6! Well, it depends what we want to say. For example, if there's no context and I say "x is a real number, x = 0" then I'm not really making a true/false statement. Whereas if I say "For all real numbers x, x = 0" or I say "There exists a real number x, x = 0" then one is clearly false and one is clearly true. Question 8 is different. Specifying that x is in Q is a much stronger condition than specifying that x is in R, because not every real number is rational. Basically this is saying: "For all real numbers x and y, then: x and y are both rational numbers (x+y) is a rational number." The initial "for all x and y in R" sets up the "universe" in which this statement is taking place, and that's a hint for the answer to that question! Hope this helps!
Yep, I got the 5 and 6. I thought of it this way too but my mind has a tendency to process the second part of a sentence, the conclusion usually, and then the reasoning but that can trip me up here. The aim to be proven makes sense, in this video it was in a vacuum but it usually wouldn't be.
HELLO HELLO HELLO!! I doubt anyone will read this because it will be buried deep down into the comments now. Now I'm brand new to set theory and mathematical logic but I did touch on logic gates when I studied Electronics over 25 years ago.. And I was wondering is there a logic gate that is capable of carrying out the Implies operation? From what I remember physical NOT gates and OR gates exist as do NAND gates (you add a NOT gate at the output to get the AND operations just as you would add a NOT gate to the output of an OR gate to get the NOR operations) Also there's the XOR (Exclusive Or) gate which gives you a 1 or True at the output only when both the inputs are different from one another, so again you can just add a NOT gate at the output and you will get the IF and only IF operations. But the Implies, I can't remember any that will perform that operation. Maybe a combination of others will?
I think I've worked it out and it was pretty simple really. You just use a standard OR gate but you connect a NOT gate to input one. So the final output will only be a 0 when input one is a 1 and input two is a 0 any other combination will give you a 1 at the output
I understand your answer now. I didn't catch on that 'or Q' meant 'OR Q' But yes that's exactly what I managed to work out using gates. Where I've said you first run input one (or P rather) through a NOT gate before it enters the OR gate you have represented that initial action using Brackets. (NOT P) OR Q. You see I'm new to all that bit, I will have to remember that one, that's important for me to know. And thank you so much
Generally I love how clear your explanations are, however in this instance, I think you missed a trick by not including : (such that). I found your list of patreon suggestions at the end very hard to parse, as commas were used for both their usual separation of statements, and to mean 'such that'.
The formal-logic _or_ rarely lines up with how it's used in plain English. Think "cake or death." If someone asks you if you'd like coffee or tea, they're not asking if you'd like both and they don't want to hear about it if you do; what they're saying is that you may have one of the two, now declare which one of the two you're choosing to have. And the idea that you don't have a preference for one over the other won't stand to scrutiny, unless you're going to sit unable to choose like Buridan's Ass. The exclusive-or is what aligns with intuition.
Absolutely love your videos. I did know most of this, learned through some very basic math courses at uni doing CS, but it's always nice with a refresher and maybe seeing things in a different light. I didn't know the uniqueness symbol, that was quite nice. However, I'm wondering about your use of comma, wouldn't a : be more correct?
I m very sad how Computer Science student go away from mathematics logic by spend time to languages and hardware ; and they don’t realize that Computer Science is created by Mathematicians.
I always loved logic notation because in a certain way I'm lazy to write in paper, so I answered these on questions on college and the math teachers and myself loved.
1. True, assuming odd/even refer to congruence to 1 and 0 (modulo 2) respectively, as those are the only possible equivalence classes modulo 2. 2. False in general, as x=0 is a real number, but 0 is neither positive nor negative. However, it can be true for x not equal to 0. 3. False, as x=0 is a counterexample again- 0 is not positive, yet 0 is also not negative. 4. True, as x=2 exists and is both prime and even. 5. False, no x exists where all y satisfy x+y=0- any y not equal to -x will break it. 6. True, for every x there does exist a y where x+y=0, specifically y=-x. 7. True, as any unequal x and y with equal squares must be negatives of each other- the only ways for squares to be equal are x=y and x=-y. 8. False, as x=sqrt(2) and y=-sqrt(2) have (x and y are rational) false, yet (x+y is rational) is still true since their sum is 0. 9. True, as any real x and y with x
I used to work with programmable logic controllers (PLC) which use "ladder logic" essentially a programming language based on relay logic. I got quite good at reading the logic and found that i could easily translate it to English in order to make sense of it to laypersons who weren't familiar with it. The plc took "inputs" that could be discreet switches or analog sensors and then based on the logic would set "outputs" that could be motor starters for examples. I thought of it as basically conditions and outcomes, if the conditions are true then make this outcome true.
@@logickedmazimoon6001 You might know this then? I was wondering is there an actual logic gate that is capable of carrying out that Implies operation? From what I remember physical NOT gates and OR gates exist in real life as does a physical NAND gate (you add a NOT gate at the output to get the AND operations just as you would add a NOT gate to the output of an OR gate to get the NOR operations). Also there's the XOR (Exclusive Or) gate which gives you a 1 (or True) at the output only when both the inputs are different from one another, so again you can just add a NOT gate at the output and you will get those IF and only IF operations like in the video which only give you a True when the inputs are the same, ie either both True or both False. But the Implies operation? I can't remember any that will perform that operation. Maybe a combination of others will?
Thank you for this great video! But what I don't understand: At 9:11 it is said: "But not multiple of four doesn't imply odd or doesn't imply even." But why do we define F->T and F->F to be true then if it is neither the first nor the second one like you said here: "But not multiple of four doesn't imply odd or doesn't imply even." "Neither nor" is not the same as "as well as", but we define it as "as well as", when we define it to be true in both cases. Or phrased differently: It is said: "Both are valid." But beeing valid or possible is not the same as beeing implied by. But we define it that way here.
I wonder whether there is a formal difference between "forall x in R: P(x)" and "x in R => P(x)" where P(x) is some proposition and R some set (e.g. the real numbers)?
(forall(x in R). P(x)) is short for (forall x. (x in R => P(x))). This is a proposition in the language and may be proven. The "matrix" (x in R => P(x)) has free variables and so can used to define a predicate in x. If you prove that expression for all x, then you've proven the former proposition. That former proposition might also be proven by other means, e.g. induction.
Edit:i just learned ∅ is a subset of all sets.. because ∅ being empty implies the lack of something. And every set contains a lack of something + something else. Which means ∅ must be a subset of all sets.( With the exception of the set that includes everything that exists and everything that doesn't) I don't understand how (12:30) a number not belonging to the empty set implies belonging to set A is true? It's only possibly true Since x is false then y must be true why? all values contained with in set A must also be present in set B. And since all values are already present in set A(since i defined it as the set of all numbere) that implies that any value present within set A is also present in set B. Since all numbers are present in both set A and Set B Set A and B are the same set. (Every value present in set B must also be present in set B because every value is present in set A, thus if there is a value that is not present in set A and only in Set B that implies the statement about set A being the set of all numbers is false.
Mathematicians really like their flipped/rotated letters. Upside-down V, rotated/flipped L, flipped A, flipped E
Mathematicians are running out of symbols and must recycle them
Isn't flipped L called Gamma Γ?
and a whole set of fancy superscript subscript fraktur serif sans serif letters that online "font generators" use to their advantage
I don't know about the A and the E, but upside down V is the very real capital Lambda from the greek Alphabet while the upside down L is the capital Gamma from the greek Alphabet. We just like greek symbols for the most part :p
I imagine that this comes from early printing - it’s much easier to rotate a block than carve a new symbol
The way my teacher explained the implication and its truth table is as follows.
Suppose I say "If I win the lottery, I will buy you a house!"
Logically, this is saying P => Q where P is "I win the lottery" and Q is "I buy you a house".
Now think about the following: in which cases are you satisfied?
When P is true and Q is true, then I kept my promise and you're happy. So T => T is T
When P is true but Q is false, then I broke my promise. I won the lottery, but didn't buy you a house. You're angry, sad, dissapointed. So T => F is F
When P is false, I haven't really made any promises. I never said what I'd do if I did NOT win the lottery. So, if I still buy you a house, you're definitely going to be happy, but even if I don't, you won't be mad because I didn't win the lottery. Hence, F=>T and F=>F are both T.
That's a nice explanation
That's similar to how I explain it to my students. Suppose I claim "If you do your homework, I'll give you an A". In which situation could you claim I lied? Only in the situation where you did your homework but I didn't give you an A.
In constructive logic, "not P" is actually expressed as "P implies falsehood", i.e., "if P, then pigs can fly."
@@markuspfeifer8473 Huh, interesting! If you know that Q is false, then P=>Q is indeed logically equivalent with ¬P
I still think my "tennis on a cylinder" idea fits better.. only one way fails to cross an imaginary line, the one where the two guys are actually playing back and forth normally, instead of around the world (which in all other cases will cross the line)
And T throws rights and F throws left. F left, because it looks more like an L with a beard.
When I was learning this, the hard part for me to digest was that implication is a logical operator. For a while I was thinking about it like "this symbol is used as equals AND an operation?!".
Nowadays, whenever I try using any complex logic with my friends I always say something like "We'll assume it's true, because it doesn't matter if it isn't", because it feels like that mindset is what "non-math" people struggle with, and many "math" people take for granted
The most helpfull comment I came across and it actually helped me finally tackle this topic.
I think the hard part is that thing we don't typically think of as true-or-false statements are so in logic/math, because everything is. Like if you write x=3, you are actually saying "the statement x = 3 is true", except the whole "is true" part is implicit. However, this is fine because setting x to a value is an intuitive enough concept you can just think of it that way. It gets weirder though with things like implications, where we especially don't think of those as true or false. Without a background in math, I feel like a lot of people would be confused if you asked "if x then y, is this true"? Once you get a grasp on the implicit things, it starts to make more sense
That "Is it a boy or a girl? Yes" joke reminded me how in the national math team training camps we do this joke all the time, some asks for example "Wait so is lunch now or do we have a lecture?" and someone else responds "Yes", man that does not get old
Having studied Maths at uni, I saw this thumbnail and thought 'YEAH?? OBVIOUSLY??' Then I actually watched the video and it's a really good video explaining the basics. Nice!
I loved the joke about "THERE EXISTS" as someone did the same for the factorial in my class some days ago 😂
More seriously, I really enjoy your videos, they're very recognizable because of their graphic identity and the music behind, and your way to show examples to be very clear, to stick little good-looking papers and to write on a black board, it's very pleasant! I particularly loved your series about foundations of numbers, but this video about logic was very good as well and I appreciated it!
Continue like that!
Having never studied this kind of math yet having years of experience in the field of programming, it's incredibly interesting seeing how a lot of concepts in both fields are equatable.
Off topic but ur protogen sona is so damn cute
How have you worked in programming for years and yet not come across prop-logic?
cute sona
@@mbdxgdb2 bro he just said " a lot of concepts in both fields are equatable.". Which mean he might have studied it in programming but not through math.
@@farhanaditya2647 Nah - you’re taught the maths before you’re taught to program if you’ve “studied it”.
I'm about 6 minutes into this video and I can't unsee the similarities(atleast so far) with logical math, and programming operators. I think I might be able to understand this.
All programming logic is inherited from traditional formal logic so there are tons of similarities!
as far as i know, boolean algebra and [0th-order] propositional logic are equivalent
Unrelated but I always watch youtube videos with auto-generated captions on, and I'm continually impressed at how far its evolved.
Especially at 2:22
Wow, I actually can't believe that!
@@AnotherRoof they've come a long way in 10 years 😄
Wowza
For every upside down A there exists a backwards E ;) In my career as a software engineer you of course lived and breathed this. You may not express it formally but logic was your constant companion.
Be careful when translating natural languages into logic: people often can be tricky.
"Everybody loves somebody" seems to have an obvious meaning. But it could either mean "There exists one person that every person loves" or "Every person has (at least) one person that they love."
(I find it fun to deliberately misinterpret ambiguous sentences.)
For all person in the set of all humans there exist another person in the set of all humans in which person loves another person.
"Every hour, some person in new york is getting run over in traffic." (What a rough time that somebody has! Getting run down every hour)
That reminds me of the Beatles song "All you need is love". People tend to hear that philosophically as something like "the only thing that any person actually needs in life is to be loved", whereas I believe the song was actually intended to mock consumerism and greed and actually meant something closer to "you already own literally everything material and/or of financial value, and now the only thing you are still lacking is love".
@@Repsack2 The Chuckle Brothers used to end their live shows by asking people to drive carefully, saying...
Paul: On your way home please take care, as statistics show that a man gets knocked down every other night of the season
Barry: Yeah, and he's getting really fed up of it now!
I wonder whether the existential symbol contains the uniqueness symbol
I'll just add this because it's something that really clarified the existential and universal quantifiers for me: the existential quantifier creates a giant OR statement, and the universal quantifier creates a giant AND statement. For example, let the universe of discourse be {0,1,2,3,4,5}. Then:
For all x: (x > 3) 0>3 and 1>3 and 2>3 and 3>3 and 4>3 and 5>3 false.
There exists x: (x > 3) 0>3 or 1>3 or 2>3 or 3>3 or 4>3 or 5>3 true.
That's a nice way of thinking about it!
Wait, there are some people that don't think of them like this?
@@quantumgaming9180 I think it was at least a year or multiple years between the time I was introduced to the quantifiers and the time I found out they were equivalent to AND statements or OR statements.
In fact there are alternative symbols for quantifiers: ⋀ and ⋁. In the same way ∏ and Σ mean product and sum over elements in a set, ⋀ and ⋁ mean conjunction and disjunction over all elements in the set.
so does that mean the Unique quantifier makes an XOR statement?
Great video! I seriously went from seeing an incomprehensible mess to thinking “well yeah, obviously”. I’m a fan of math, but never felt I had enough talent to go get an advanced degree in it. But your videos make these esoteric sounding ideas easy to grasp. I would love it if you covered Gödel’s incompleteness theorem at some point. Love your work!
Hey! You're tricking me into studying maths by making interesting and well explained videos! Not cool!
please keep making them thank you
Wow I finally get the implication part. Looking at 11:06,
If P is a circle inside Q in this 'space of all possibilities'
then you can point you finger at any point on that space and say:
Point x is inside P and Q
Point x is outside of P but inside of Q
Point x is outside of P and outside of Q
But you cant point to a space that is inside of P and outside of Q. Thats why that is F, its an impossible state.
In the case of "greg is a cat->greg is a mammal" the only contradiction is where greg is a cat and not a mammal. No other scenario is contradictictory.
I think your reciprocal proposition is actually a really strong argument for why implication is the way that it is.
For all real numbers x, (if x is not 0, then there exists a real number y such that xy=1)
We really want the implication to be true for all x for our universal quantifier, but there is a value of x where the first statement of the implication is false. The thing we're trying to prove doesn't really care what happens when x=0, but we still need the implication as a whole to be true for all x, including 0, the thing we were trying to exclude. So, we just define that case to be true no matter what, because it means we don't have to worry about it breaking our quantifier.
Isn't there already a 'such that' symbol?
Never understood those weird math symbols but this video really helped.
Also as a Software Developer I can find many similarites in the language of maths and code.
When I was in undergrad, all CS majors had to double in something else and most chose math.
@dootie8285 Underrated comment!
Where was this guy when I was doing my maths degree?!? Really clearly explained
0:10 definition of the multiplicative inverse. Booya.
My cousin used to always answer questions like that. “Do you want to watch Frozen or Moana”
“Yes.”
“Would you like water or milk”
“Yes. Yes I would”
Very good video!
I think that the solution to the exercise at the end of the video is this
1: true (ironically this is the only one i'm unsure about) 2: false (because of 0) 3: false (because of 0) 4: true (because there's 2) 5: false (there isn't a value that works for every y) 6: true (for every x there is that works) 7: true (i think this doesn't need explanation) 8: false (because there are numbers that aren't elements of Q and their sum is an element of Q: π and -π if you sum them you get 0 which is an element of Q 9: true (because R is dense) 10: true (the only value is 0)
Could you pls clarify no.5 a bit more
@@ziadhossamelden9241 there isn’t a value that added to any y equals 0
The proportion says that a value that works with any of the real numbers but there isn’t because for examples for 3 only -3 works such as 3+(-3)=0 but it doesn’t work for -4
I was looking for this comment, I have a question, tho:
1: Does this mean we all think that 0 is even? It follows the pattern, but it's just kind of weird lol.
And I got 8 and 10 wrong. Both were obviously you're answer after thinking about them further. I didn't think about transcendentals for 8, and I didn't think about how the uniqueness of 0 is the special characteristic that makes statement 10 true.
@@kindlinEvery even number "x" is a multiple of 2, which means you can write it as x=2k where k is an integer. 0 is obviously an integer and 2*0=0 => 0 is an even number.
I believe 10 is false. Edit: I was wrong
This video is delightful. Mathematics would probably be more palatable to a general audience, if children were given a helpful introduction to logic in the early days. Most of the benefit of mathematics in real life is logic. Many people who struggle with math either fail to see its relevance or fail to grasp the basic logic underpinning the statements.
1. True. All integers are either odd or even.
This is a direct consequence of the Theorem of Euclidean Division, which states:
For every pair of integers m,n, there exists a unique pair of integers q,r, with r < n, such that m = qn + r.
In this case, n = 2; therefore, all integers can be expressed as either 2n or 2n + 1.
2. False. There exists a real number, namely 0, such that it is neither positive nor negative. This follows axiomatically from the fact that the set of real numbers is an ordered field.
3. False. There exists a real number, namely 0, such that 0 is not positive, but 0 is not negative. This is equivalent to the previous proposition.
4. True. There exists a natural number, namely 2, such that 2 is prime and also even.
2 is prime beacuse it cannot be expressed as a product of two smaller natural numbers. Being the first natural number greater than 1, the only possible "product of two smaller natural numbers" is 1x1 = 1, not 2.
2 is even because it can be expressed as 2 = 2x1, that is, 2 = 2n for n=1.
5. False. There exists NO real number which has the property of "destroying all numbers" through addition, that is, that the result of it added to any number will always result in 0.
To prove this, suppose, by contradiction, that such a number x exists. That is, x + y = 0, for any real number y. Then, take y + 1: x + (y + 1) = (x + y) + 1 = 0 + 1 = 1 =/= 0, which is a contradiction.
(On the other hand, there exists such a number for multiplication: 0 "destroys all numbers" through multiplication since y.0 = 0, for any real y.)
6. True. For every real number x, there exists a real number y, called the "additive inverse" of x, with the property that x + y = 0. This number is y = -x. This is a property that defines the set of the real numbers as a field.
7. True. To prove this, consider the second member of the "and" relation: x² = y². By subtracting y², we have x² - y² = 0. Factoring, we have (x - y)(x + y) = 0. A product of two real numbers is zero if, and only if, one of the numbers is zero. Therefore, either x - y = 0, which would mean x = y (not allowed by our premise), or x + y = 0, which would mean x = -y. Therefore, the implication holds.
8. False.
P: x is rational and y is rational.
Q: (x+y) is rational.
Q does not imply P: this means that, if (x+y) is rational, then x and y need not be both rational.
In fact, for x = √2 and y = -√2, we have (x+y) = √2 - √2 = 0 rational, but neither x nor y is rational.
9. True.
In fact, we can take z = (x+y)/2 which has the required property:
z - x = (x+y)/2 - x = (y-x)/2 which is positive when x < y, meaning x < z.
y - z = y - (x+y)/2 = (y-x)/2 which is positive when x < y, meaning z < y.
10. True. That unique number is 0.
It is true that 0 has the required property, since for y > 0, 0² = 0 < y.
The proof that 0 is the unique number with this property is as follows:
Suppose, by contradiction, that another number x =/= 0 has the same property.
Then, x² > 0, which implies x²/2 > 0. Take y = x²/2. y > 0 but x² > y = x²/2, which is a contradiction.
(This proof requires the knowledge of the fact that: x² = 0 iff x = 0, x² > 0 otherwise.)
I'm not convinced by 7. What if x and y are both 0?
Oh, it's in the proposition that x and y are unequal.
Beautiful list! I also have a question about 7, but for a different reason. I believe that for all real numbers x,y, if x = -y, then x ≠ y and x² = y² is also true, meaning it has a two way relationship. Is it a problem if one way implication is stated for a two way relationship like that? After typing it, my gut is saying that (a b) ==> (a ==> b) (I hope I did that right 😅), but I am unsure. Do you know the answer to this?
I also need to ask if 0 counts as an even number for the response to question 1. I believe it is a counterexample to the statement.
you're so smart. Can we be friends?
I was always curious about logic notation. Now I won't have to be haunted with pages and pages of unknown symbols when I choose to study this subject. Very good video!
22:28 another way to prove that one false would be, since x is an element of the real numbers, and y covers all the real numbers, y also covers x, and x is never less than itself
26:37
1. true (either have to be true)
2. true (same as last)
3. true (not positives are negative, but the sign could also be )
4. false (no prime is even since even is divisible by 2)
5. false (**BOTH** **MUST** be 0, and Ay e R is not only 0)
6. false (same as last)
7. true (the only number squared that isnt itself is its negative)
8. true (both x and y must be in Q for the x+y to be in Q) (EDIT: false, irrational+irrational can equal a rational, e.g. π+(1-π)=1)
9. true (you can always fit a real number between 2 other)
10. false (no one REAL number squared is < 0)
@@Nick12_45 not bad! Check Q8 though...
At first,i used to find logic math/logic philosophy hard because i thought that it was for prodigies or geniuses,well basically,i'm good at math,but i was not that good at logic math,i only knew the element of,and the sets,but because of you, I'm starting to love logic math,and it got easier for me,and I'm starting to get hooked up with it,thank you😊.
A major factor in the confusion of the or statement is the implied use in natural language of "or" as "exclusively or". With the drinks example, I don't know if I'd be happy being served two hot drinks. Those kinda have a time limit.
I'm a tenth grader and I understand the notation. Before I actually watch the video, here is how I would say that example statement:
"For every real x that is nonzero, there exists a real y such that xy=1"
I thank my math teacher for teaching at above grade level, and we also strangely learned formal logic and its notation in philosophy classes 👍
Wait did the joke at 6:21 just go over my head, or does UK English keyboard layout actually have the logical negation symbol on it? I get the feeling I missed the joke but it would be cool if that symbol was actually standard on a keyboard
I didn't realise this was only a UK thing! But yeah our key below Esc and to the left of 1 as the ¬ symbol on it. As well as ` which only TeX-savvy people use. And also ¦ which... your guess is as good as mine.
@@AnotherRoof Oh interesting! Yeah, on an US English keyboard layout, that key has ` (backtick) also, but the other symbol on it, which you reach with shift, is ~ (tilde). But you say there's a third symbol there? I recognize that broken-vertical-bar but the US layout only has the regular vertical bar | on the same key as the backslash (under backspace)
After looking it up the US and UK keyboard layouts are MUCH more different than I thought! Dang.
@@Scum42 Yeah I realise that now -- we also have the vertical bar on the backslash key but it's left of Z. I do remember some of the differences now because I once bought a US-layout keyboard but I like our big, tall 'enter' key, But I never realised ¬ was a UK thing. Don't know why -- literally no idea what it's used for outside of formal logic!
Having that key be the negation/inversion symbol would make so much more sense in vim. Because that key inverts capitalization.
Lovely video! My exposure to logic has been in computer programming, really neat to see the parallels!!
edit: my logic answers
1) If x is in the set of Integers Z, the Odd set holds x or the Even set holds x.
This is True (assuming 0 has parity)
2) If x is in the set of Real Numbers R, the Positive set holds x or the Negative set holds x.
False (assuming 0 is Real and Unsigned)
3) For every value x contained in the set of Real Numbers R, if Positive doesn't hold x then Negative does hold x
False (assuming 0 is Real and Unsigned)
4) There exists some value x in the set of Natural Numbers N where x is prime or x is even
True (This will work for any prime or even number)
5) There exists some value x in the set of Real Numbers R, which for every value y in the set of Real Numbers x+y=0
False (by contradiction: x:4+ y:3 ≠ 0)
6) For every value x contained in the set of Real Numbers R, there exists some value y contained by R in which x+y=0
True (Any number's opposite added to the same number will yield 0)
7) For any two values x,y contained by the set of Real Numbers R, if x ≠ y AND the square of x is the square of y, that x = -y
True (if x and y are not equal but their squares are the same, then the magnitude of x and y must be identical. X^2 = Y^2, X = ±Y)
8) For any two values x,y contained by the set of Real Numbers R, that if x and y are both contained in the set of Determinate Fractions Q, the sum of x and y must also be contained in the set of Determinate Fractions Q (and vice versa)
True (the sum of two fractional numbers will never yield a non-determinate fractional number, and the addends of a fractional number will always be two determinate fractional numbers--otherwise the definition of a detemrinate fractional number breaks)
9) For any two values x,y contained by the set of Real Numbers R, if x < y, then there exists some number z contained by the set of Real Numbers R that falls between x and y.
True (Real Numbers allows non-wholes, and a non-whole number can __always__ be subtracted or added to. An easy way to guarantee this is by picking the minimum place value of x and y together, making it one order of magnitude smaller, and adding a single unit of that place value to x)
10) There exists a unique value x contained in the set of Real Numbers R, that with any value y contained in the set of Real Numbers R, wherein if y > 0, the square of x will be less than y
False (by contradiction: More than one value. x:2^2 < y:5 and x:2^2 < y:6, therefore x:2 is not unique)
Super super fun brain teasers!!!!! My favorite was number 8 and I do hope I'm correct on these. Thanks for a fantastic video.
edit edit: somebody added an irrational number [(π) that can be a determinate fraction] to itself on #8 and proved me wrong. Cheers!!!
For #4, you used the wrong connective; properly it should be "There exists a natural number x such that x is prime and x is even." Which is true, x=2.
For #5, you're correct, but you your proof is not a proof by contradiction, and isn't sufficient to prove the statement true or false, because a claim is being made about a property of all real numbers. A proper proof by contradiction here would be something like y = 1-x, x + (1-x) = 1, 1!= 0.
For #8, even considering your edit, the rationale is wrong. Q is the set of rational numbers; π is not a rational number, nor is π + π, so plugging it in for x or y creates a vacuous statement and doesn't prove anything. A better example would the counterexample x=π, y = 1-π. x+y=1, which is in Q, but neither (x in Q and y in Q) does not hold, so the biconditional is not satisfied and the statement is false.
For #9, you're correct, but I think a better explanation is that it's possible to define z such that the value of z always falls between x and y; the simplest example I can think of is z = (x+y)/2.
For #10, you're misinterpreting the meaning of ∀. "∀y ∈ R" means that the proposition must be true for all values of y, not for any single value of y. The statement is true; x=0 is the unique value whose square is smaller than any positive number.
I think the best way to think of ∀ and ∃ is that in both cases, you must consider every possible value of the variable. For ∀x, the predicate must be true for every possible value of x, but for ∃y, you only have to prove that out of every single possible value of y, the predicate is true for at least one.
I would try to avoid using "any" in phrases like "for any value" because that usage is ambiguous; "for any value" could mean that we should be able to plug any conceivable value in and the statement is true, or it could mean that we want it to be true given at of the set. For example "x+1=2" is true for "any" real number because it's true for 1, but it's not true for "any" real number because it's not true for 2.
Also for #8, another way to prove it false is that if you let (x ∧ y) = 1/2, then x + y = 1 which is not within the set of rational numbers. Likely, if x + y ∈ Q, then it doesn’t mean that (x ∧ y) ∈ Q because you can let x = 1 and y = 1/2 which means x + y = 3/2, and even tho the sum is rational, its components x and y are not. 🙏
@@zaydsalcedo3009 All integers are rational numbers. For example, 2 can be written as 2/1
Using mathematical symbolism and logic can provide a powerful bridge to connect theological/metaphysical concepts with scientific/physical descriptions in a rigorous way. Instead of relying solely on binary true/false valuations, engaging non-contradictory/contradictory modes of reasoning could be highly fruitful.
Here are some thoughts on how we could apply this approach:
1. Multi-valued and Fuzzy Logics
Rather than classical bivalent logic, we could explore multi-valued algebraic logic systems that allow for more nuanced truth valuations beyond just 0 and 1. This could capture theological notions of paradox, ineffability, and transcendent reality that goes beyond strict binarization.
Fuzzy logics which admit truth values in the continuous range [0,1] could model metaphysical concepts that are irreducibly vague or context-dependent. Non-contradictory/contradictory could then be represented by sub-ranges of the multi-valued domain.
2. Paraconsistent Logics
Paraconsistent logical systems are designed to deal with contradictions in a controlled, discriminating way rather than just admitting logical explosion. This could allow rigorously reasoning about metaphysical statements that are paradoxical or logically inconsistent from a classical perspective.
Non-explosive paraconsistent frameworks like relevance logic could formalize theological ideas involving prescribed inconsistencies or contradictories without trivializing the entire system. Non-contradictory and contradictory conditions could be encoded precisely.
3. Modal Logics and Intuitionistic Systems
Modal logics explicitly capture notions of necessity, possibility, and ontological modalities. We could use graded/fuzzy modal systems to represent transcendent, ineffable realities beyond typical ontological constraints.
Intuitionistic logics based on constructive reasoning avoid strict bivalence and the principle of excluded middle. This could model metaphysical concepts that are not straightforwardly decidable in a binary fashion.
4. Substructural Logics and Resource Semantics
Substructural logics like linear logic impose resource-consciousness by controlling structural rules like weakening and contraction. This limited, pay-as-you-go approach could capture theological ideas of existential scarcity, ontological austerity, and irreducible indeterminacies.
Phase semantics and resource models in these logics could provide novel metaphysical interpretations and construct ontological stances beyond strictly bivalent modes.
5. Topological Semantics and Cohesion
Cohesive topological models using homotopy theory and algebraic topological semantics could provide a powerful geometric metaphor for non-contradictory/contradictory conditions in terms of intrinsic continuities, boundaries, and points of inflection.
This could unify metaphysical and scientific descriptions by embedding them in a common topological setting where contradictions are smoothly navigable via continuous pathways rather than pure bivalence.
By leveraging the immense richness of mathematical logic and non-classical reasoning frameworks, we could indeed use symbolic representations to bridge theological abstractions and physical observations in a philosophically robust yet scientifically grounded manner.
The non-contradictory/contradictory mode could become a new conceptual lens, expanding rigid true/false binaries into a continuum of coherence where metaphysics and science fluidly intersect. I'm happy to further explore concrete examples of how to apply these ideas to specific theological/metaphysical notions and their scientific counterparts.
Is this chat gpt?
In language, we tend to use "or" to mean "xor" or "exclusive or". This version is true when one of the inputs is true, but not both. This is why the "yes" to "or" questions play as jokes.
Actually, if I say 'x or y' without context, it's ambiguous. 'either x or y' is xor and 'x and/or y' is 'or'
@@Anonymous-df8it
No, I would have to say, “either x or y, but not both,” not just “either x or y.”
However, we often do say “or” when we mean “xor.”
Do you want steak or chicken? Yes, both, please.
@@bethhentges a) Why wouldn't "either x or y" be sufficient? b) "However, we often do say “or” when we mean “xor.”" Your example question isn't meant to be taken literally; even if you interpret the or as xor, you still don't get the intended meaning (see 'is it a boy or a girl?')
@@Anonymous-df8it
Could be intersex! Both.
Dear another roof,
You may discourage me all you want,
in no way possible will that ever stop
me from consuming your carefully
crafted content.
Weirdly enough, a few days ago, I was thinking in bed "How would I introduce the concepts of Boolean logic to a middle school/high school class?" (I am totally serious) Your video tracked almost precisely with the way I would have laid it out (I didn't go into quantifiers, but I did cover a few things like DeMorgan's Laws, and various alternate notations). otoh, you taught me something I did not know (or at least remember?), the uniqueness quantifier ∃!
Turning the lights on by flipping the wall switch, and then turning them off.
1+1=0
Compare two circuits: one in series, one in parallel. Then put a switch in each circuit.
I know us French people tend to make everything different when it comes to maths, but I just wanted to tell you that for us, 0 is always a natural number and 1 is never prime, except if you wanna define it otherwise but I've never seen anyone do so yet
1 is not prime in America either
For some reason in the USA, in K-12 ed, and the first two years of college, we make a distinction between natural numbers (positive integers) and whole numbers (non-negative integers).
Then once you are in your third yr at college and start group theory/abstract algebra, then we change the definition of natural number to include zero.
In the USA, the number written
-3 is “negative three,” NOT “minus three.” The word “minus” should be used only for the operation of subtraction. In everyday life, we often hear “minus” used incorrectly as “negative.”
Also, in the USA -3 is an integer, but it’s not a whole number, because the whole numbers are the non-negative integers only.
I tell my students that definitions develop over time. They start as a general description, and they get more precise as the object becomes more understood. Along the way, “edge cases” are sometimes included and other times not. It’s important to know what those edge cases are so that when you engage with a new person/course/text, you will know you need to agree as to whether or not the definition is inclusive of the edge case or not.
For the purpose of the new discussion we need to know:
Is zero a natural number?
Can a line be parallel to itself?
Is a rectangle a trapezoid?
When we say suppose a and b are two _____ , are we allowing them to be the same _____ , or are we assuming they are distinct?
Regardless of which choice we make, we need to keep that in mind as we go forward in the statements of new theorems and definitions.
So glad to hear I'm not the only one who remembers the AND symbol as the n in fish n' chips (it's also how i remember the difference between union and intersection in set theory)
For the cover: (btw I’m 12 but still understands this due to knowing Python and JavaScript)
If any number (variable x) that is real number and not zero, there will be at least one real number (variable y) that when multiplied to x, result becomes 1.
Example:
x = 5; y = 0.2; xy = 1
x = 0.01; y = 100; xy = 1
x = -255; y = -255; xy = 1
x = pi; y = 1/pi; xy = 1
x = 1.23e+300; y = 1.23e-300; xy = 1
x = cos(0); y = cos(0); xy = 1
x = 69420; y = 1/69420; xy = 1
Invalid because they’re not real numbers or is 0
x = infinity
x = i
x = 0
27:03 funny how "and/or" was part of the ending.
I guess outside mathematical logic "or" is more commonly understood as "xor".
If we apply that to the frase it could be "and + xor", or just "or".
From a boolean point if view, I think "or" is enough, but to comply with common practice for language and communication it is better to use "and/or"
If you want to convey xor, it's better to say either... or... imo. 'or' is ambiguous
@@Anonymous-df8it yes 'exclusive or' is fairly easy to express in words.
"Ether A or B" is very specific
(aka. "xor" in boolean algebra).
Similarly apply for and.
"A and B" is very specific.
(aka. "and" in boolean algebra).
My primary focus was possibly was lost (in cyberspace), because of other matters matters mentioned in the short message; and it's difficult to convey what's emphasized as primary focus in text.
I can't come up with a short and umabigious way to say 'inclusive or' in words; to prevent misunderstandings.
"A or B or both" is on way, but I think it's longer than I want.
"A and/or B" is short, but feels 'messy'.
@@thorbjrnhellehaven5766 May you please elaborate on what you mean by 'messy'?
@@Anonymous-df8it The slash is often read as "or" when used with other words, but I think saying "A and-or-or B" is kind of a messy and kind or confusing tongue twister..
Of course you can say "A and-or B", and that's possibly the way most people would say it.
'messy' as in cluttered with different things, because it's difficult to use only words (plain lerrets), but easier to convey by mixing in special characters
@@thorbjrnhellehaven5766 Most would say the latter
"by short I mean there's four videos in the series, it is three and a half hours long, but, you know"
I really like your jokes!
Great! That's exactly what I'm learning and being tested on right now
This is the part of maths that actually feels like learning a language, and being able to just translate it into an English sentence is very satisfying
haven't finished the video yet but im trying to apply what i know so far by trying to define the XOR operation:
R XOR Q = (R∨Q)∧￢(R∧Q)
reasoning: in XOR, one of R and Q has to be true (the first term) and they cannot be both true (the second)
watching more, i guess you could also define:
R XOR Q = ￢(R⇔Q)
I dont have this keys on the keyboard so I use ! for not, & for and, | for or as you'l see in programming
the first way is the formal along with (!R&Q) | (R&!Q)
the second way is just a reflaction of the fact that "if and only if" means they are the same which is what the xnor gate checks, and xor is not xnor
but xor also have a diffrent symbol which is a + in a circle
OMG you're so helpful your video is really interesting and informative. Also love the jokes ,it break the tense that built with in me every time the problem and the materials getting difficult. Ty kind sir. You're Video is Amazing all love from me you sure put a lot of care into it♥🥰.
The way I would explain implication is basically that, because of the law of the excluded middle, P implies Q still has to have a truth value when P is false. Either way could work, but saying implication is true when P is false has less weird implications. It really should be "indeterminate", but that's not an option.
This is an amazing basis for logic, solving, and all the frontier forms of programming/gaming
For all the programmers watching this video marvelling at the similarity with various programming languages, that is no accident. Programming languages were basically conceived as a way to implement first order logic.
0:40, well it might not have been geared for me, but somehow I had never come across that use of “!” to mean “a unique” before, so I learned something! :)
Laughing at how when I saw this first I paused the video, read it, and said “oh yeah, obviously” 😂
omg thanks for explaining this in such an understandable way... a lot of these ideas I already kinda knew from functional programming concepts (the "for all" and "there exists" seemed very familiar like the .all() and .any() methods for iterators in Rust) but I had no clue how people describe them in math terms. Especially that "implies" part was sorta tricky, but that circle diagram was pretty helpful.
if statement {
assert!(implication);
}
I have almost finished my CS degree and never understood the implication and this guy managed to explain it to me, what a guy, implication is one of the most important "tools" in math an it made me always feel insecure because i've never understood it correctly, thank you very much
Definitely gave me a clearer perspective on why implication works the way it does. Thank you!
'so long and/or farewell'
since 'and v or' has the same truth table as or, and equivalent statement should be
'so long or farewell'
ps not sure if my language is mathematically rigorous or not. if error lemme know tq tq
I appreciate how you explain everything in detail!
man I love all these videos so much, can't wait to see what's next in store, I'm super curious
Fantastic video! I'm super curious to see where this series goes next!
The way you explained this topic was funny 😂 and I liked it
Thank you for making this hard topic look simple and interesting
Can you do a video explaining all 16 diagrams of VENN-diagrams, in relation to Binary numbers and logical reasoning.
So, I would like to entirely understand why the Venn diagrams are the best way to use for logical reasoning.
Thank you very much in advance.
I would read this “for all x in R it is true that there exists a y in R such that x times y is equal to 1”
What a wonderful video. Subscribed.
1. true (idk how to prove it tho)
2. false. counterexample: 0
3. false. counterexample: 0
4. true. for example: 2
5. false. counterexample: y=-x-1
6. true. y=-x, therefore x+y=x+(-x)=x-x=0
7. true. (-y)^2=y^2
8. false. counterexample: x=pi, y=4-pi
9. true. for example: (x+y)/2
10. true. that number is 0
5:40 hhh no cuz i genuinely laughed! wonderful lesson!
This video is very nice for beginners, I'll probably recommend this video to students at the beginning of a Math proving class so they know what they'll be getting into.
"Well, who doesn't like pi?"
*vihart has entered the chat*
nice one
24:08 wouldn't the opposite of the inside statement also be true?
if there's a real number y such that xy = 1, then x must not be 0.
Wouldn't that be an "if, and only if" statement then?
Great video and such a good explanation. This will help me to understand some of my subjects on engineering
Thank you very much for this! Stumbled upon this randomly and I always wanted to know yet i never had the ambition to really look it up
Your video is perfect! Don't listen to these robot comments. Thank you for your time and effort! I am learning a lot! You are appreciated!
there exists an x which is a real number such that if x is not equal to zero then there is a reciprocal of x
I'm 33 seconds in but i remember my formal logic course from first semester uni.
This says:
"For all x which are an element of the real numbers, if x is not equal to zero, then it follows that there exists a y which is element of the real numbers, such that x times y is equal to 1."
The upside down A fittingly stands for "for _all"_ and the mirrored E stands for "there _exists"_ .
For those who want an easy translation, the intro statement says “for all x that exist in the real numbers, so long as x does not equal zero, then there exists a number y which when multiplied to create xy, you get 1.”
Maybe the context should be read better as “you can find a number y such that xy equals 1.”
My attempt at the 10 questions:
1. True, all integers are either odd or even.
2. False, 0 is a rational that is neither positive or negative.
3. False, again 0 is not positive and also not negative.
4. True, 2 is prime and even.
5. False, there is no number that is 0 when anything is added to it.
6. True, y=-x
7. True, I can't think of any counterexamples.
8. False, there can exists two non-rational numbers that add to a rational number, (pi)+(1-pi)=1
9. True, you can take the average of x and y to get z.
10. True, x=0.
Nice job! 👍
I'd just like to add the following note:
Just because you are unaware of any counterexamples doesn't immediately imply that the statement must be true. Though, luckily for you, #7 can be proven to be True because if x² = y² then x² - y² = 0. Applying the difference of squares and solving for the cases where x ≠ y gives us the desired x = -y.
Hope this helps, and stay curious! :)
10 is wrong because when x = 0, y =0 will not work
As the relatively fames phrase goes "One man's modus ponens is another man's modus tollens, another man's disjunctive syllogism or another man's indirect proof.".
Since these four logical expressions are logically equavalent to each other:
[A⇒B] ≡ [¬B⇒¬A] ≡ [¬A∨B] ≡ ¬[A∧¬B]
the corresponding syllogisms are then also logically equivalent to each other:
(P1) A⇒B
(P2) A
(C1) B [from P1 and P2 by modus ponens]
(P3) ¬B⇒¬A [from P1 by contraposition]
(C2) B [from P2 and P3 by modus ponens]
(P4) ¬A∨B [from P1 by material implication]
(C3) B [from P2 and P4 by disjunctive syllogism]
(P5) ¬[A∧¬B] [from P4 by de Morgan's law]
(P6.0) ¬B [indirect proof assumption]
(P6.1) A∧¬B [from P2 and P6.0 by conjunction introduction]
(P6.2) [A∧¬B]∧¬[A∧¬B] [from P5 and P6.1 by conjunction introduction]
(C4) B [from P2, P5 and P6.0-P6.2 by indirect proof]
So I guess, that these are all from the same partition of syllogisms, which we might call the simplest valid derivation from two premises.
Just Another Roof for you.😉
4:22 I remember the v is OR because it looks like the set union symbol lol.
I keep thinking that implication is just an artifact of the premise that truth conditions can only be either True or False. Functionally, A -> B is equivalent to ¬A V B, which is easier to interpret. Though the arrow makes for a nice shorthand.
Considering the alternatives, saying an implication is false when the condition in not met just gives you conjunction. Any other alternative gives you the identities of the inputs. But realistically, I think what people would _want_ to say is that an implication whose condition isn't met is Unknown "U", rather than True "T". That's just not an option, though.
Say one wanted to represent A -> B with sets. They could draw A as a set subsumed by B, but that carries the subtle additional idea that A exists. It's the same discrepancy with "The king of France is bald." Does that mean that the king of France _exists_ and is bald, or only the implicational relationship?
Similarly, What is the probability of B given A, if the probability of A is 0? Freaky stuff...
What’s the probability of B given A if I know the probability of A is zero?
Thanks. I love that example.
Since implications are themselves either true or false, does that mean we can nest them? For example
( P => Q ) => (¬Q => ¬P )
which should be true for every proposition P and Q
Absolutely!
Yes,
P=>Q
and
~Q=>~P
are logically equivalent.
They are either both true xor both false.
They are contrapositives of each other.
Spending most of my time in programming and some in digital logic my first thought where is exclusive or.
The end problems, 5 is true but 6 is false right?
Also what is the distinction between just stating x belongs to R and using for all x belongs to R? In logical statements, the're treated the same right? Getting this inquiry from the 8th question in the end slate, it seems redundant to have specified R when you then go on to specify Q
Check your answers for 5 and 6!
Well, it depends what we want to say. For example, if there's no context and I say "x is a real number, x = 0" then I'm not really making a true/false statement. Whereas if I say "For all real numbers x, x = 0" or I say "There exists a real number x, x = 0" then one is clearly false and one is clearly true.
Question 8 is different. Specifying that x is in Q is a much stronger condition than specifying that x is in R, because not every real number is rational. Basically this is saying:
"For all real numbers x and y, then: x and y are both rational numbers (x+y) is a rational number."
The initial "for all x and y in R" sets up the "universe" in which this statement is taking place, and that's a hint for the answer to that question!
Hope this helps!
Yep, I got the 5 and 6. I thought of it this way too but my mind has a tendency to process the second part of a sentence, the conclusion usually, and then the reasoning but that can trip me up here.
The aim to be proven makes sense, in this video it was in a vacuum but it usually wouldn't be.
HELLO HELLO HELLO!! I doubt anyone will read this because it will be buried deep down into the comments now. Now I'm brand new to set theory and mathematical logic but I did touch on logic gates when I studied Electronics over 25 years ago..
And I was wondering is there a logic gate that is capable of carrying out the Implies operation? From what I remember physical NOT gates and OR gates exist as do NAND gates (you add a NOT gate at the output to get the AND operations just as you would add a NOT gate to the output of an OR gate to get the NOR operations) Also there's the XOR (Exclusive Or) gate which gives you a 1 or True at the output only when both the inputs are different from one another, so again you can just add a NOT gate at the output and you will get the IF and only IF operations.
But the Implies, I can't remember any that will perform that operation. Maybe a combination of others will?
"P implies Q" is logically equivalent to "(not P) or Q". Draw up truth tables of both to convince yourself!
Thanks for watching!
I think I've worked it out and it was pretty simple really. You just use a standard OR gate but you connect a NOT gate to input one. So the final output will only be a 0 when input one is a 1 and input two is a 0 any other combination will give you a 1 at the output
I understand your answer now. I didn't catch on that 'or Q' meant 'OR Q' But yes that's exactly what I managed to work out using gates.
Where I've said you first run input one (or P rather) through a NOT gate before it enters the OR gate you have represented that initial action using Brackets. (NOT P) OR Q.
You see I'm new to all that bit, I will have to remember that one, that's important for me to know.
And thank you so much
Generally I love how clear your explanations are, however in this instance, I think you missed a trick by not including : (such that). I found your list of patreon suggestions at the end very hard to parse, as commas were used for both their usual separation of statements, and to mean 'such that'.
The formal-logic _or_ rarely lines up with how it's used in plain English. Think "cake or death." If someone asks you if you'd like coffee or tea, they're not asking if you'd like both and they don't want to hear about it if you do; what they're saying is that you may have one of the two, now declare which one of the two you're choosing to have. And the idea that you don't have a preference for one over the other won't stand to scrutiny, unless you're going to sit unable to choose like Buridan's Ass. The exclusive-or is what aligns with intuition.
I was always ... curious about symbolic logic, thanks for clarifying!
Absolutely love your videos. I did know most of this, learned through some very basic math courses at uni doing CS, but it's always nice with a refresher and maybe seeing things in a different light. I didn't know the uniqueness symbol, that was quite nice. However, I'm wondering about your use of comma, wouldn't a : be more correct?
These videos are really interesting; I'm curious to see what you do next.
A good short recap on my first weeks lectures as a maths undergrad.
Ive already done my final exam in calculus, linear algebra and discrete maths. But I wish I know about this channel sooner. 😹
There's something weirdly attractive about some dude successfully teaching me mathematical bureaucratese under an hour.
"Be or not to be" - bv~b - tautology
Doesn't it just say multiplicitive inverses exist for all nonzero real numbers?
I m very sad how Computer Science student go away from mathematics logic by spend time to languages and hardware ; and they don’t realize that Computer Science is created by Mathematicians.
I always loved logic notation because in a certain way I'm lazy to write in paper, so I answered these on questions on college and the math teachers and myself loved.
1. True, assuming odd/even refer to congruence to 1 and 0 (modulo 2) respectively, as those are the only possible equivalence classes modulo 2.
2. False in general, as x=0 is a real number, but 0 is neither positive nor negative. However, it can be true for x not equal to 0.
3. False, as x=0 is a counterexample again- 0 is not positive, yet 0 is also not negative.
4. True, as x=2 exists and is both prime and even.
5. False, no x exists where all y satisfy x+y=0- any y not equal to -x will break it.
6. True, for every x there does exist a y where x+y=0, specifically y=-x.
7. True, as any unequal x and y with equal squares must be negatives of each other- the only ways for squares to be equal are x=y and x=-y.
8. False, as x=sqrt(2) and y=-sqrt(2) have (x and y are rational) false, yet (x+y is rational) is still true since their sum is 0.
9. True, as any real x and y with x
I used to work with programmable logic controllers (PLC) which use "ladder logic" essentially a programming language based on relay logic. I got quite good at reading the logic and found that i could easily translate it to English in order to make sense of it to laypersons who weren't familiar with it. The plc took "inputs" that could be discreet switches or analog sensors and then based on the logic would set "outputs" that could be motor starters for examples. I thought of it as basically conditions and outcomes, if the conditions are true then make this outcome true.
I think you'd really like logic gates and truth tables and boolean algebra, take a look!
@@logickedmazimoon6001 You might know this then? I was wondering is there an actual logic gate that is capable of carrying out that Implies operation? From what I remember physical NOT gates and OR gates exist in real life as does a physical NAND gate (you add a NOT gate at the output to get the AND operations just as you would add a NOT gate to the output of an OR gate to get the NOR operations).
Also there's the XOR (Exclusive Or) gate which gives you a 1 (or True) at the output only when both the inputs are different from one another, so again you can just add a NOT gate at the output and you will get those IF and only IF operations like in the video which only give you a True when the inputs are the same, ie either both True or both False.
But the Implies operation? I can't remember any that will perform that operation. Maybe a combination of others will?
@@harryedwards4080 Theres no dedicated implies function/gate but you can make one using NAND and NOT gates
Very clear, thanks !
Thank you for this great video!
But what I don't understand: At 9:11 it is said: "But not multiple of four doesn't imply odd or doesn't imply even."
But why do we define F->T and F->F to be true then if it is neither the first nor the second one like you said here: "But not multiple of four doesn't imply odd or doesn't imply even."
"Neither nor" is not the same as "as well as", but we define it as "as well as", when we define it to be true in both cases.
Or phrased differently: It is said: "Both are valid." But beeing valid or possible is not the same as beeing implied by. But we define it that way here.
it's really "could imply", and that "T -> F" is the only thing that could refute it, but it's just called "imply"
@@notwithouttext I have never heard that it means "could imply". Where did you find this?
@@vincentv.3992 just my guess take my thing with grain of salt
@@notwithouttext Ok, thank you anyway.:)
I wonder whether there is a formal difference between "forall x in R: P(x)" and "x in R => P(x)" where P(x) is some proposition and R some set (e.g. the real numbers)?
(forall(x in R). P(x)) is short for (forall x. (x in R => P(x))). This is a proposition in the language and may be proven. The "matrix" (x in R => P(x)) has free variables and so can used to define a predicate in x. If you prove that expression for all x, then you've proven the former proposition. That former proposition might also be proven by other means, e.g. induction.
The first says “for all x in R such that P(x) is true.”
The second says that “for all x in R, P(x) is true.”
They are not the same.
Edit:i just learned ∅ is a subset of all sets.. because ∅ being empty implies the lack of something. And every set contains a lack of something + something else. Which means ∅ must be a subset of all sets.( With the exception of the set that includes everything that exists and everything that doesn't)
I don't understand how (12:30) a number not belonging to the empty set implies belonging to set A is true?
It's only possibly true
Since x is false then y must be true why? all values contained with in set A must also be present in set B. And since all values are already present in set A(since i defined it as the set of all numbere) that implies that any value present within set A is also present in set B. Since all numbers are present in both set A and Set B
Set A and B are the same set. (Every value present in set B must also be present in set B because every value is present in set A, thus if there is a value that is not present in set A and only in Set B that implies the statement about set A being the set of all numbers is false.
xor is missing => outrageous imperfection!))
Also it's a bit confusing that in propositions "=" may be false unlike in equasions
This is such a beautiful video. Thanks so much for making it. I really appreciate it dude ❤
Awesome explanation, thank you!