As a mathematician, I have a couple of remarks. I've never seen the single line arrow --> used to denote implication. In every paper I've seen, double arrows are exclusively used for that shorthand. Single line arrows are instead reserved for functional notation, like "f: A --> B" to denote "f is a function from A to B," or for limiting behavior, like "f(x) --> 0 as x--> 0." I'm surprised that you didn't mention the subset symbol. We use that all the time. When we use \in in our written proofs, we always symbolically define the set on the right hand side. So we wouldn't use "Even #," but rather, say, 2Z. We also don't use symbols as often as a first-year proofs class would have you believe. I'm much more likely to say "For every" instead of the \forall symbol, and "There is a unique" instead of "\exists !". These symbols are used more commonly on blackboards and in lectures, to get the idea out more quickly. I will use the implication symbol ==> sometimes, but I use it as a replacement for the word "implies." And when I do this, it's usually in an effort to state the implication without explicitly committing to the hypothesis. I might say something like, "P ==> Q, which trivially implies what we want, so we may assume that P is false." I also only do this when P and Q are already symbolically heavy statements, e.g., "x \in Z ==> x^2 \in Z." Contradiction symbols aren't really present in papers, but on blackboards, I've taken to writing ==>
Note that some of these symbols are precisely defined mathematical objects (like the Implication), but other symbols (such that, therefore) are semantical values that exist to make the proof readable for a human, but they are not themselves mathematical objects.
exactly as a romanian i couldn't for the life of me figure out what s.t. meant before he actually said what it means in romania this same notation is called a.î. meaning "astfel încât" which translates to "such that"
Thanks @ki for sharing! I have seen the two arrows touching point-to-point before but not the diagonal hashtag symbol. I'll include this on a future video coming out next week on the topic of Proof by Contrapositive.
There's really not enough established convention to distinguish between these two senses. You will need to clarify with the author. As with Xracess, I know of a context (in the Isabelle/HOL proof assistant system) where => is used at in the context of theorems implying other theorems, and -> is used as a binary operation in logical formulae.
ST can also mean Subject To. It pops up in optimization problems. For instance maximize some function subject to the sum of the independent variables equals X.
Upside-down T means this is a contradiction, and regular T means something like "this is a true / logical statement". I like the upside-down triangle of dots, which mean "because".
One symbol my Math teacher loves using is : before =. := stands for "is defined as" Say, for example, the definition of a function f f : |R* -> |R x |-> y := f(x) := (x²+1)/x Or, to be more succinct f : |R* -> |R x |-> y := (x²+1)/x Or f : |R* -> |R x |-> (x²+1)/x
0:20 often in math "symbols are used to do some operations" So too in truth functional logic sirm Even in the predicate calculus, the implication arrow means to do a calculation (unless they are in strings transformed by rewrite rules.) And that arrow does not tightly cohere with the english 'implies'.
Whilst the meaning is different to that of the word 'implies' in regular english, i believe A⇒B is spoken as A implies B, and ⇒ is referred to as the 'implies sign', even if it doesn't really mean 'implies' in the usual sense.
@@josephthomas4900 Indeed. I believe logicians call ot "material implication". It means simply "(not A) or B". Which is not to say other logics are impossible. But this is standard.
There existS a word which means what you're trying to say. "There, exist.", is a command, of sorts, but, "There exist symbols", is nonsense, because "exist" is a verb, with no subject, but "exists" is a quality, kinda like a preposition. I am not a professional English teacher.
What is the point of standing in front of written mathematical narrative and explaining it to baffled learners? Use overhead projector or long pointer to explain away finer points of ready solutions.
It's usage tends to be pretty specific, it's often the sort of thing where you'll prove a lemma, then have the logic after it where you can reference already showing the lemma(or cite it from elsewhere) therefore step where we use the lemma without going through the intermediary steps. So it's very much a "step 1, step 2, invoke lemma therefore step 6, step 7, step 8" style progress. For the most part it's not nice, it's better to actually use the steps from the lemma/invoked proof if it's not well known rather than citing it, but it does get used.
And don’t take me wrong, being lazy while keeping the work effective (in this case communication) is great and it is the core principle that drives us to a better world
This was super important and really well done -- except for your self-deprecation, which was not funny and a total waste of time. You're a teacher! Get on with it!
You talk too much , you flap your hands you could take off. You should have prepared your talk so your words are minimal . A boring explanation of set theory notation with some silly comments thrown in for good measure. Sorry to be brutal but if you are going to make videos you need to be clear and not tell the audience they know when clearly it is your role to tell them. You decided to be the teacher.
He touched on this in the video, but there are two main reasons: First, it’s much quicker to write. It might be hard to appreciate this if you’re someone who doesn’t write proofs often, but having to say “for any real numbers a and b there exists a third real number c” over and over again is super annoying and cumbersome compared to “∀ a, b ∈ ℝ ∃ c ∈ ℝ”. Again, writing it out once is reasonable, but if you need to say the same thing three times in a proof, and do five or so proofs in a homework assignment, you very quickly come to appreciate how these symbols shorten things. The second reason is that unlike English words, these symbols correspond directly to logical concepts. My favorite example of this is the word “is”, which is used in English to mean equality (the king of Camelot is Arthur Pendragon) but also sometimes to mean element inclusion (Arthur Pendragon is a king) and also sometimes to mean set inclusion (a king is a monarch). Mathematical notation lets you distinguish between these: (Arthur Pendragon) = (king of Camelot) is the first statement, (Arthur Pendragon) ∈ King is the second statement, and King ⊆ Monarch is the third statement. Different meanings get different symbols so there is no possibility for confusion.
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As a mathematician, I have a couple of remarks.
I've never seen the single line arrow --> used to denote implication. In every paper I've seen, double arrows are exclusively used for that shorthand. Single line arrows are instead reserved for functional notation, like "f: A --> B" to denote "f is a function from A to B," or for limiting behavior, like "f(x) --> 0 as x--> 0."
I'm surprised that you didn't mention the subset symbol. We use that all the time.
When we use \in in our written proofs, we always symbolically define the set on the right hand side. So we wouldn't use "Even #," but rather, say, 2Z.
We also don't use symbols as often as a first-year proofs class would have you believe. I'm much more likely to say "For every" instead of the \forall symbol, and "There is a unique" instead of "\exists !". These symbols are used more commonly on blackboards and in lectures, to get the idea out more quickly.
I will use the implication symbol ==> sometimes, but I use it as a replacement for the word "implies." And when I do this, it's usually in an effort to state the implication without explicitly committing to the hypothesis. I might say something like, "P ==> Q, which trivially implies what we want, so we may assume that P is false." I also only do this when P and Q are already symbolically heavy statements, e.g., "x \in Z ==> x^2 \in Z."
Contradiction symbols aren't really present in papers, but on blackboards, I've taken to writing ==>
Note that some of these symbols are precisely defined mathematical objects (like the Implication), but other symbols (such that, therefore) are semantical values that exist to make the proof readable for a human, but they are not themselves mathematical objects.
exactly
as a romanian i couldn't for the life of me figure out what s.t. meant before he actually said what it means
in romania this same notation is called a.î. meaning "astfel încât" which translates to "such that"
@@greengreen110 dude nobody cares about your country
I've seen the contradiction symbol written as two arrows touching point-to-point (like this: -->
Thanks @ki for sharing! I have seen the two arrows touching point-to-point before but not the diagonal hashtag symbol. I'll include this on a future video coming out next week on the topic of Proof by Contrapositive.
they look so much more complicated than they really are
"Abnormally simple" I love that phrase.
The two symbols for implies are similar hut have sifferent meanings. One on the left is "proven to imply" the other is "claimed to imply".
also you use the left one for statements and use the right one for formulas
There's really not enough established convention to distinguish between these two senses. You will need to clarify with the author. As with Xracess, I know of a context (in the Isabelle/HOL proof assistant system) where => is used at in the context of theorems implying other theorems, and -> is used as a binary operation in logical formulae.
Both "⇒" and "→" are used for implies in logic. Although the first one is more common, there usually is not a distinction between the two.
ST can also mean Subject To. It pops up in optimization problems. For instance maximize some function subject to the sum of the independent variables equals X.
Upside-down T means this is a contradiction, and regular T means something like "this is a true / logical statement".
I like the upside-down triangle of dots, which mean "because".
I've also seen contradiction by ->
Upside down T means "perpendicular" in my mathematical dialect.
@@ExplosiveBrohoof That too, in geometry.
Thank you for this video, I found it really helpful and enjoyed it!
One symbol my Math teacher loves using is : before =.
:= stands for "is defined as"
Say, for example, the definition of a function f
f : |R* -> |R
x |-> y := f(x) := (x²+1)/x
Or, to be more succinct
f : |R* -> |R
x |-> y := (x²+1)/x
Or
f : |R* -> |R
x |-> (x²+1)/x
Huh, I've been using it to mean 'is reassigned to' for iterative stuff. I suppose I really should just be subscripting.
@@pauld9690 "≔" is commonly used for assignment in algorithmic contexts, so its fine to use it that way
For "implies both ways", i.f.f. (or iff ?) can also be used, which stands for "if and only if".
@17:12 some people use
fact: .'. is therefore
and '.' is because
Thanks and thank the commenters.
Excellent video. Thank you!
really underrated
Missed "QED", or square symbol.
You are confusing symbols that clearly don't have the same exact meaning. Beware of the *types* .
hello teacher
i really love math
but i feel its hard
so i want to learn from zero
do you have the first lesson of math?
I prefer the colon : for s.t.
sir respect from J&K
Another common one is iff. which stands for "if and only if". iff. and bidirection implication () are logically equal.
0:20 often in math "symbols are used to do some operations"
So too in truth functional logic sirm Even in the predicate calculus, the implication arrow means to do a calculation (unless they are in strings transformed by rewrite rules.)
And that arrow does not tightly cohere with the english 'implies'.
Whilst the meaning is different to that of the word 'implies' in regular english, i believe A⇒B is spoken as A implies B, and ⇒ is referred to as the 'implies sign', even if it doesn't really mean 'implies' in the usual sense.
@@josephthomas4900 Indeed. I believe logicians call ot "material implication". It means simply "(not A) or B".
Which is not to say other logics are impossible. But this is standard.
By the way sometimes ':' is used for such that.
I was under the impression that the double-headed arrow meant "if and only if".
It does. "If and only if" means the same thing as "implies both ways."
Amazingly clear. Thank you so much!
I NEEDED THIS VIDEO SO MUCH
Very good video, thanks!
Therefore? But there are only three
dy/dx IS A FRACTION of infinitesimals !!!!
Ur very good .. U helped me ur awesome
This is an amazing video
Thank you so much
Has anyone ever written a comprehensive dictionary or encyclopedia of mathematical symbols.
The closest thing to this is the book "Principia Mathematica"
↯ Is the symbol I've seen used for contradiction most of the time.
Great video.
There existS a word which means what you're trying to say.
"There, exist.", is a command, of sorts, but, "There exist symbols", is nonsense, because "exist" is a verb, with no subject, but "exists" is a quality, kinda like a preposition.
I am not a professional English teacher.
What is the point of standing in front of written mathematical narrative and explaining it to baffled learners? Use overhead projector or long pointer to explain away finer points of ready solutions.
symbols plural
0:26 GG
✓°→√Ω¶{×÷}[]≤≥⟩⟨%±-·ⁿ⅒
He'll I am student from India
You can’t have everything that’s in front of your face.
Pretty much nobody uses the "three dot triangle" symbol. I have a phd in math, and I've never ever seen anyone use that symbol.
It's usage tends to be pretty specific, it's often the sort of thing where you'll prove a lemma, then have the logic after it where you can reference already showing the lemma(or cite it from elsewhere) therefore step where we use the lemma without going through the intermediary steps.
So it's very much a "step 1, step 2, invoke lemma therefore step 6, step 7, step 8" style progress. For the most part it's not nice, it's better to actually use the steps from the lemma/invoked proof if it's not well known rather than citing it, but it does get used.
My junior high algebra teacher used it, so I keep it handy for my own notes.
I've seen it a shit ton, maybe one of our experiences are just skewed
very common in engineering
💚 Super nerd! 😅💚
f(x)!?!?!?
p and q are co-prime. I would express it as "GCF(p, q) = 1"
This is the proof that mathematicians are lazy af😂
And don’t take me wrong, being lazy while keeping the work effective (in this case communication) is great and it is the core principle that drives us to a better world
This was super important and really well done -- except for your self-deprecation, which was not funny and a total waste of time. You're a teacher! Get on with it!
graphic thumbnail, graphic logo; but video on a physical board - no thanks.
Would you like your money back?
What's wrong with that?
Don't forget to wash your hands on the way out. Bye.
Fr was a bit disapointed
I know right man! I mean like, every other thumbnail on this website is a FLAWLESS representation of the video that they're on!
You talk too much , you flap your hands you could take off. You should have prepared your talk so your words are minimal . A boring explanation of set theory notation with some silly comments thrown in for good measure. Sorry to be brutal but if you are going to make videos you need to be clear and not tell the audience they know when clearly it is your role to tell them. You decided to be the teacher.
Why not just write words. I don like when people overcomplicate with maths for no reason at all.
Like mfs who "prove 1 + 1 = 2"
He touched on this in the video, but there are two main reasons:
First, it’s much quicker to write. It might be hard to appreciate this if you’re someone who doesn’t write proofs often, but having to say “for any real numbers a and b there exists a third real number c” over and over again is super annoying and cumbersome compared to “∀ a, b ∈ ℝ ∃ c ∈ ℝ”. Again, writing it out once is reasonable, but if you need to say the same thing three times in a proof, and do five or so proofs in a homework assignment, you very quickly come to appreciate how these symbols shorten things.
The second reason is that unlike English words, these symbols correspond directly to logical concepts. My favorite example of this is the word “is”, which is used in English to mean equality (the king of Camelot is Arthur Pendragon) but also sometimes to mean element inclusion (Arthur Pendragon is a king) and also sometimes to mean set inclusion (a king is a monarch). Mathematical notation lets you distinguish between these: (Arthur Pendragon) = (king of Camelot) is the first statement, (Arthur Pendragon) ∈ King is the second statement, and King ⊆ Monarch is the third statement. Different meanings get different symbols so there is no possibility for confusion.
Thanks 👍👍👍
⊥⊥⊥ could also represent a contradiction