This course might have mistakes or define things slightly differently than you will/have seen in other texts. If you find an error please reply to this comment with timestamp. I will do my best to make corrections (in the video description or otherwise) with the tools youtube allows at the time.
1:00:00 The negation of a conjunction of two formulae is the disjunction of the negated formulae. That is ¬ (m odd ∧ n odd) is logically equivalent to (¬m odd ∨ ¬n odd). That's a logically weaker statement than (m even ∧ n even).
thank you for speaking VERY slowly and defining everything. Your pacing is excellent, and i LOVE that you constantly say "so what does this mean?". You are a true teacher who wants students to reflect and take a step back to digest and it is a very underrated skillset as an educator. A lot of math professors move too fast and automatically expect students to be at their level. They are great mathematicians first and educators second, which can make things greatly intimidating and disheartening for some students. Keep up the GREAT work!
You can also buy books those are things too. Also in a course you get feedback directly from the instructor so its not really the same but still this is a great video which should always be complementary with a book.
not really the same thing. a face to face in class is way better, instructors responses you instantly when you ask during classes. you try to follow through the exercises, they can check them for you, feedback you, give you advices.
Lectures aren’t worth shit. I always learned best from textbooks and forum discussions on them. But I’m autistic so maybe you can extract meaning from other people’s words.
Hey man, I'm a freshman at Stanford studying math right now and this video has been super helpful! Just wanted to let you know that you're making an impact in my life (and likely my mathematical career). Thanks for this amazing video!
This is a great review. We did all of these techniques in my discrete math class, which was a really good comprehensive introduction to mathematical proofs.
10:30 That is what puts most people off: proofs that have been brewing for 2500 years in the heads of human kind are presented in the most slick way and as straight forward. I think one needs to make a huge disclaimer around that. Usually maths student start to prove the laws of set operations, because it is plugging in definitions. The genious spark comes later, after having seen A LOT of ways how to prove things.
Thank you for this course. I'm gonna rewatch this multiple times before getting out my pencil and paper to let it sink in and essentially prepping my brain so that I can eventually write proofs. My goal is to learn advanced computer algorithms and write these algorithms in python. This video is my first step.
Hello teacher Thank you so much for your interesting lessons, i do appreciate your job. I wish you peace and happiness under the sky of prosperity. All the best. Take care and have a good time.
I had "french typed" math education, and I really don't recall anything about the naming: Modus Ponens and Modus Tollens. While we studied the logical statements p-》q is equivalent to not q -》p. In programming, this helps simplify some complex logical expressions. Very useful indeed.
ultra-cool af math proving video. tyvm for me bringing back to the tautologies. i used to have a table or listing of all tautologies. they were fun to use in programming and proofs.
Math proofs can feel intimidating at first, but once I started breaking them down step-by-step, they became much clearer. A solid foundation is key, and having structured resources like SolutionInn’s study modules really helped me build confidence in tackling proofs efficiently.
I self studied proof by a book proof in math by richard hammack was a while ago so might have messed up the name but ur video is a goldmine for any new student .
hello sir, thank you for your tutorial. May I ask why 5:46 there is a subtle and important definition of rational number which is m/n is a reduced fraction? I have googled the definition of rational number but it only states that "Any number that can be expressed in the form 𝑝/𝑞, where 𝑝 and 𝑞 are integers,"
If we had q - something is a square and p - it is a rectangle then it wouldn't work. If something is a square it is also a rectangle, not the other way. Modus Tollens, (12:20) Anyway, really useful course.
So I should do this course, for a big secret, astounded at the similarities between what you need to prepare for a successful unique experiment and how you prove in mathematics, this is all knew to me. Got me thinking properly what is an irrational number, I am sure I will enjoy it.
yep. An irrational number is one that's already in its most reduced form. So, the fact that m and n had a common factor of 2 that means it was not in its reduced form which is a contradiction therefore sqrt(2) must be irrational.
Be careful, in English the structure either-or IS exclusive. Btw, this was a great course, I really enjoyed it. Also, in Hungary 0 is considered a natural number.
Why can't we notice that if n=2a+1 (odd) and then square to get n²=4a²+4a+1=2(2a²+2)+1, which is also odd and write this proof in a formal way @56:40 ? Update: I see that this doesn't prove that all odd n² have odd n. There might be an even n that squares to odd. I know there isn't, but I think the my given solution above does not disprove it, so the proof is incomplete.
5:21 I think you should first prove that square root of 2 exists in the set of real numbers in the first place To do this, you should define real numbers using cauchy sequences and come up with a cauchy sequence that converges to a number whose square is 2.
Would you say for #1:35:44, you can multiply the subcase 2 with -1 and it will be the same case except the x is now negative instead of the y because of W.L.O.G it still goes either and it works like Subcase 1? I think in my case it complicates it more if this holds true.
This is so helpful!! I had a request though, can you make a video on strong mathematical tools which can be used in higher problems of physics, I was doing one problem involving black body which used riemann zeta function. That video will be really helpful
@@florentikaa I'm 45 almost now. Talking to peole in everyday life is hard because I don't know how regular people use words when they speak 😂😂😂 a blessing and a curse
1:00:54 I think you have a mistake the Negations of m and n odd is the either of them is even but not necessarily both of them. Regardless I wanted to thank you for the video even though I'm familiar with the concepts you explain them in a very good way.
59:57 it's wrong, the negation of "m and n odd" is "m or n is even". You even showed it in 32:51 A correct proof: Assume without loss of generalities that m is even then, m=2k for some k. Therefore mn=2kn (an even number) => mn is even
Would it be possible to copy timestamps for when each episode starts into the description? That would let RUclips divide it into chapters, making things easier to follow. This is great but it would be better with the chapter divisions.
But where did you use the assumption "M over N is a reduced fraction" in your first proof? I mean for it to be denied you need to use it some where in the proof at first don't you? For me, you prooved that the square root of two is a rational number that can be written as a not reduced yet fraction Please can someone explain so i can continue the video?
Ive always had a problem with the first proof about proving irrationality. What if instead of sprt2 you just out sqrt4 there instead. Wouldnt that just prove that 2 (sqrt4) is also irrational?
2:18:22 There are no "false proofs". Speaking of a "false" proof (or a "true" proof for that matter) is committing a category error. Proofs are either valid or invalid. None of them are true, false, sweet or yellow.
Im here because my German university doesn’t offer an intro to proofs class at all. They throw us into Real Analysis in the first semester and just assume we know all this logic stuff.
Because we see that since m^2=2n^2, m is twice another integer, the definition of an even number. Finding n to be an even number as well comes from the fact that m is even, as he showed, and is the contradiction.
I have a proof but unfortunately RUclips doesn't give me enough characters to write it here. Also, if I were to reveal the proof, were would the fun be?
This course might have mistakes or define things slightly differently than you will/have seen in other texts. If you find an error please reply to this comment with timestamp. I will do my best to make corrections (in the video description or otherwise) with the tools youtube allows at the time.
At 59:55, the negation should be “m OR n are even”, applying de Morgan’s law, right?
@@christianpio1261 exactly that's what I was about to write.
Thanks 👍
there's a typo of exists at 35:43
there's a typo of absolute at 39:10
1:00:00 The negation of a conjunction of two formulae is the disjunction of the negated formulae. That is ¬ (m odd ∧ n odd) is logically equivalent to (¬m odd ∨ ¬n odd). That's a logically weaker statement than (m even ∧ n even).
thank you for speaking VERY slowly and defining everything. Your pacing is excellent, and i LOVE that you constantly say "so what does this mean?". You are a true teacher who wants students to reflect and take a step back to digest and it is a very underrated skillset as an educator.
A lot of math professors move too fast and automatically expect students to be at their level. They are great mathematicians first and educators second, which can make things greatly intimidating and disheartening for some students. Keep up the GREAT work!
Was going to say this. We appreciate the pace!
0:00 What's a Proof
10:44 Logical Rules
17:38 Mathematical Sets
30:38 Quantifiers
41:50 Direct Proofs
52:35 Contrapositive
1:01:39 "If and Only If"
1:12:28 Proof by Contradiction
1:23:01 Theorems are always true.
1:27:48 Proof by Cases (Exhaustion)
1:38:56 Mathematical Induction
1:51:17 Strong Induction
1:54:40 Introduction to Function.
2:01:49 Existence Proofs
2:11:01 Uniqueness Proofs
2:18:22 False Proofs
.
god’s work 👏🏽👏🏽👏🏽
Thanks so much!
1:09:50 proof by contradiction
colleges are charging thousands of dollars for a course like this while this guy is doing it for free
You can also buy books those are things too. Also in a course you get feedback directly from the instructor so its not really the same but still this is a great video which should always be complementary with a book.
I'd recommend checking out Book of Proof, which has a free online version
@@sumofat4994exactly…
not really the same thing. a face to face in class is way better, instructors responses you instantly when you ask during classes. you try to follow through the exercises, they can check them for you, feedback you, give you advices.
Lectures aren’t worth shit. I always learned best from textbooks and forum discussions on them. But I’m autistic so maybe you can extract meaning from other people’s words.
I’m 17 and we are doing this at school so this video was a great help with learning the basics
I wish I watched this video 25 years ago when I was in college.
Your explanation is very easy and clear.
I love you, man. Thanls for the almost 2h30min. I'm gonna do a very good use of them, don't worry.
this is the video that i have been looking since my high school days . it's a hidden gem 💎💎💎
Hey man, I'm a freshman at Stanford studying math right now and this video has been super helpful! Just wanted to let you know that you're making an impact in my life (and likely my mathematical career). Thanks for this amazing video!
Wow, I have recommended this video to my discrete mathematics course students... That's very helpful and to the point. ❤
Why do I like this so much? Why does math work better for my brain with proofs?
thank you for eliminating barriers to access for this information 🎊
Your presentation style is very engaging, especially for non-native speakers, as your slow speech makes it easier for them to follow.
I dont even have words to thank you. You are great.
This is a great review. We did all of these techniques in my discrete math class, which was a really good comprehensive introduction to mathematical proofs.
I wish I could superlike this. This is great. Thank you for speaking slowly.
Yep, im gonna be a math major. Thanks, def helped me with figuring it out
10:30 That is what puts most people off: proofs that have been brewing for 2500 years in the heads of human kind are presented in the most slick way and as straight forward. I think one needs to make a huge disclaimer around that.
Usually maths student start to prove the laws of set operations, because it is plugging in definitions. The genious spark comes later, after having seen A LOT of ways how to prove things.
Thank you for this course. I'm gonna rewatch this multiple times before getting out my pencil and paper to let it sink in and essentially prepping my brain so that I can eventually write proofs. My goal is to learn advanced computer algorithms and write these algorithms in python. This video is my first step.
The ultimate best math lesson on RUclips. I wish I knew even a part of it, while learning for my BSc. Thank you very much.
THIS IS GOLD
Thanks!
Welcome! Thank you!
Thx man. I'm studying systems engineering and I'm seeing the discrete Math course. This helps a lot.
This is exactly what I was looking for to review and prep for the fall. Thank you for the video
This is awesome - it really could've helped me when I was taking modern analysis in college!
Hello teacher
Thank you so much for your interesting lessons, i do appreciate your job.
I wish you peace and happiness under the sky of prosperity.
All the best.
Take care and have a good time.
Love you bro! I wish I had found this before taking honours math classes in first year.
I had "french typed" math education, and I really don't recall anything about the naming: Modus Ponens and Modus Tollens. While we studied the logical statements p-》q is equivalent to not q -》p. In programming, this helps simplify some complex logical expressions. Very useful indeed.
I recently just thought of learning proofs, and it is here in my recommendations...
Alright, I subscribed.
Why is your pfp upside down
ultra-cool af math proving video. tyvm for me bringing back to the tautologies. i used to have a table or listing of all tautologies. they were fun to use in programming and proofs.
FINALLY SOMEONE GIVING PROOFS LOVE U BROOOO
Suppose √2 is in Q
√2 = a/b
2 = a²/b²
2b² = a² -> a = 2c
2b² = (2c)²
2b² = 4c²
b² = 2c² -> b = 2d
(2d)² = 2c²
One must imagine Sisyphus happy
This isn't correct: 2b² = a² implies a = 2c
sqrt(2b²) = sqrt(2) sqrt(b²) = sqrt(a²)
implies
a = sqrt(2) c
Math proofs can feel intimidating at first, but once I started breaking them down step-by-step, they became much clearer. A solid foundation is key, and having structured resources like SolutionInn’s study modules really helped me build confidence in tackling proofs efficiently.
A gem!
I self studied proof by a book proof in math by richard hammack was a while ago so might have messed up the name but ur video is a goldmine for any new student .
Thank you so much. Your proofs are clearly presented.
I picked up proofs a week ago the timing was so good
hello sir, thank you for your tutorial.
May I ask why 5:46 there is a subtle and important definition of rational number which is m/n is a reduced fraction?
I have googled the definition of rational number but it only states that "Any number that can be expressed in the form 𝑝/𝑞, where 𝑝 and 𝑞 are integers,"
If we had q - something is a square and p - it is a rectangle then it wouldn't work. If something is a square it is also a rectangle, not the other way. Modus Tollens, (12:20)
Anyway, really useful course.
So I should do this course, for a big secret, astounded at the similarities between what you need to prepare for a successful unique experiment and how you prove in mathematics, this is all knew to me. Got me thinking properly what is an irrational number, I am sure I will enjoy it.
yep. An irrational number is one that's already in its most reduced form. So, the fact that m and n had a common factor of 2 that means it was not in its reduced form which is a contradiction therefore sqrt(2) must be irrational.
Be careful, in English the structure either-or IS exclusive. Btw, this was a great course, I really enjoyed it.
Also, in Hungary 0 is considered a natural number.
Good Sir, THIS, this what I'm looking for!❤
Why can't we notice that if n=2a+1 (odd) and then square to get n²=4a²+4a+1=2(2a²+2)+1, which is also odd and write this proof in a formal way @56:40 ?
Update: I see that this doesn't prove that all odd n² have odd n. There might be an even n that squares to odd. I know there isn't, but I think the my given solution above does not disprove it, so the proof is incomplete.
Nice video and this course should be offered at college level.
5:21
I think you should first prove that square root of 2 exists in the set of real numbers in the first place
To do this, you should define real numbers using cauchy sequences and come up with a cauchy sequence that converges to a number whose square is 2.
3:44 - Typo @ "Exisits"
5:13 - Typo @ "connot"
I love, love, ❤❤ this!!!
You are the best youtube teacher I have ever Seem. Love from India ❤
Would you say for #1:35:44, you can multiply the subcase 2 with -1 and it will be the same case except the x is now negative instead of the y because of W.L.O.G it still goes either and it works like Subcase 1? I think in my case it complicates it more if this holds true.
This is so helpful!! I had a request though, can you make a video on strong mathematical tools which can be used in higher problems of physics, I was doing one problem involving black body which used riemann zeta function. That video will be really helpful
@59:55 the negation should be m odd or n odd, not m odd and n odd
1.5x speed works well 😂 On a serious note, great video!
Please make more comprehensive videos like this on various maths topic even if they are hours long
the transitions between chapters were well done lol
14:58 I remember my dad trying to use "non-exclusive or" to explain "or" in regular language skills to me as a kid 😂😂
oh wow, how its working on you?
@@florentikaa I'm 45 almost now. Talking to peole in everyday life is hard because I don't know how regular people use words when they speak 😂😂😂 a blessing and a curse
1:00:54
I think you have a mistake
the Negations of m and n odd is the either of them is even but not necessarily both of them.
Regardless I wanted to thank you for the video even though I'm familiar with the concepts you explain them in a very good way.
Bro is serving to mankind. Brother really, thanks❤
can u make a course like this for all international maths olympiad main topics. Such as combinatrics, algebra, geometry, number theory and other stuff
After a breakup i lost my purpose in life , still sad but math gave me a purpose
59:57 it's wrong, the negation of "m and n odd" is "m or n is even". You even showed it in 32:51
A correct proof:
Assume without loss of generalities that m is even then, m=2k for some k.
Therefore mn=2kn (an even number) => mn is even
Would it be possible to copy timestamps for when each episode starts into the description? That would let RUclips divide it into chapters, making things easier to follow.
This is great but it would be better with the chapter divisions.
Thanks a lot .
But where did you use the assumption "M over N is a reduced fraction" in your first proof?
I mean for it to be denied you need to use it some where in the proof at first don't you?
For me, you prooved that the square root of two is a rational number that can be written as a not reduced yet fraction
Please can someone explain so i can continue the video?
What is this sorcery? Thanks❤ I was just searching on my PC how to prepare for math olympiads (wanna go to the IMO) and this popped up. ❤❤❤
gl with the IMO!! im planning to go to the IPhO sometime and its nice to see another olympiad person here :)
@@pinkshortcomedy gl to you as well!
Merci beaucoup ❤
Ive always had a problem with the first proof about proving irrationality. What if instead of sprt2 you just out sqrt4 there instead. Wouldnt that just prove that 2 (sqrt4) is also irrational?
4|m² does not imply 4|m.
Booooom 💪😎❤️
How the hell did you just get m is not equal to n in 55:27
2:18:22 There are no "false proofs". Speaking of a "false" proof (or a "true" proof for that matter) is committing a category error. Proofs are either valid or invalid. None of them are true, false, sweet or yellow.
Im here because my German university doesn’t offer an intro to proofs class at all. They throw us into Real Analysis in the first semester and just assume we know all this logic stuff.
13:27
Just nitpicking
Here, the relation of q and r is not strong, maybe it would be better to say “People wear boots”
Thank you very much
Continue )))
Oh, these are from the udemy course :)
btw do you plan to solve those interesting or difficult integrals again? they were pretty fun!
That's the plan!
dope
Can you make a video about defining x^A
Where A is a matrix
can't a and b be the same number here? 1:51:00
I just can’t fathom the contrapositiv proof… how do they correlate to each other??
Thanks 👍
11:15 Conditionals are not implications!
Imma download this
36:38 can anybody explain why the ST shifted its place in negation..
timestamps for myself:
october 7th 2024: 17:45
In first proof why cant m an n be even intergers meaning sqrt(2) is rational?
Because we see that since m^2=2n^2, m is twice another integer, the definition of an even number. Finding n to be an even number as well comes from the fact that m is even, as he showed, and is the contradiction.
Dope Bro
POV: you bought the course on udemy and now find it free on RUclips💀
You can request a refund:)
@@BriTheMathGuyudemy has a proof that refund Is irrational
@@BriTheMathGuy its literally my class 12 maths syllabus
Thankyou
From india ❤
Will this course present a proof of the Riemann Hypothesis?
I have a proof but unfortunately RUclips doesn't give me enough characters to write it here. Also, if I were to reveal the proof, were would the fun be?
11:31 LMAO AHGAHGAGAGAJHAHA LMAO..... kmai
Hi have u guys read jay cunning long form proof , any opinion?
You misspelled “argument” in the first slide buddy…just a heads up
QED
👏👏👍
Спасибки❤
>no timestamps
>choosing beggar
@12:32
why do i always wake up to this????
Hello 1st Comment
Mathematics lover ❤
why does this video have just 5k views
is it my imagination or did u say "taken 4 granite" instead of "taken 4 granted"?
Way to make a simple proof complicated