An Introduction to Mathematical Proofs
ฝัง
- เผยแพร่เมื่อ 27 มี.ค. 2024
- ✍️ This video will give you a basic understanding of how Mathematical Proofs work and what Mathematics University Students usually struggle with at the beginning of their academic career. Now also with (cheap) Subtitles ‼️‼️‼️
❗ A bit of a more beginner oriented video this time around, but one that felt important to me personally. As always: thanks for the patience. ❤
🎶 Massive thank you and shoutouts to the creator of Baba is You - Arvi Teikari for letting me use these fantastic tracks of the game's OST ❤:
- Wall Is Stop - The beginning
- Cog Is Push - Solitary Isle
🎵 other music can be found here:
pastebin.com/VTyn3SEu
Thank you, Mr Percent! I can finally mathematically prove that 1 = 7!
Hate to break it to you, but 1 != 5040
I can prove that 6 = 3!
@@broorI can prove that 3!= 7
Assume that the numbers 1 and 7 indicate a summation of vectors, whose starting and ending points are same in both summations.
Using the vector theorem, if you start the vectors from the end point of the nose of the number 1 until its foot, you get the summed vector that looks more or less like this - _\_
Again, use vector theorem for the number 7 taking the direction starting from its nose to its foot. We see that the sum is _\_
Both are facing the same direction. If the nose of number 1 is tilted downwards, the length may not be the same between both 1 and 7. However if its straight maybe we can then say its equal.
Here we proved that 1 = 7, or _\_ = _\_ , which also means 1 || 7.
@@broor I approve of factorial
What you define as a "proof by contradiction" is typically known as a "proof by contra-positive",
from formal logic if A -> B then it must also be true that ~B -> ~A.
Usually a "proof by contradiction" is when we need to prove A->B so we assume (A wedge ~B)
and reach a contradiction (a statement that is always false like (C wedge ~C), and thus A->B is proven.
Ah true, we did actually call this "proof by contra-positive" the same thing at uni but i see what you mean. But i guess then this negation of an implication is what i wanted to teach here anyways, just gave it a bit of a wrong name.
Thanks for paying such close attention to detail!
@@0prcent No problem, I found you from your Jordan curve video, truly remarkable content
@@0prcent _"How to prove anything"?_ Quite the ambitious thumbnail, though imo out of reach for the introduction. Good 3b1b animation is not a cure-all silver bullet, after all.
@@ultimaxkom8728 The title is obviously hyperbole.
@@ethanbottomley-mason8447 Yes, it's clickbait.
got a math olympiad tomorrow and somehow qualified without understanding proofs, but now i feel like im ready. thanks mate
Thanks for explaining the "if and only if" part.
Echoing other commenters, the "proof by contradiction" you showed is actually a "proof by contrapositive." That said, proof by contrapositive is often very helpful. I almost always start with some direct approach, and if that's too difficult, I'll look at the contrapositive. If I get lucky and the contrapositive is easier, I'll write that out, then rewrite it as a direct proof to make it look "cleaner."
Aside: Something that's often overlooked is how much reading proofs improves your proof writing skills. I can't count how many problems I've solved by just applying the overall structure/approach someone else used.
there is some magical power in youtube algorithm which suggests me top tier quality mathematical video about topic which is relevant in school with no mistakes nor any delay
thank you so much!! (and thanks for the algorithm of course)
baba is you music is a perfect match for this subject
Can you prove the riemann hypothesis in a next video? Thanks!
Yeah this would be really helpful 🙏🙏
@@SlightSmile very would be an understatement
Really helpful for my financial status
This is a very important video. I and many other people struggle with proofs.
Thank you.
I never even had this introductuin in my course. Even though this is my second year doing computer science this was very helpfull. Thank you!
Thanks for covering this.
wow, i really like your editing. the music choice, the graphics, the effects... very comfy. makes me want to study, even
aw thanks!
Incredible man, keep it up.
nice music choice
Baba is baba and not you
Baba is win?@@user-sk4kg4hr3k
@@user-sk4kg4hr3k now prove by contradiction (contra-positive)
Really good video, thanks!
thank u
good video!, also i love the use of the va11halla OST :3
thx! so happy i got to use it in this video :)
oooh that's what it is I was digging it the entire time
the full music list is also in the description btw if youre curious about any of the other songs
baba is logic
baba
Les go you also recognised the bg music
Well made video, here's an algorithm boost!
this is such a good video. If this is what proofs are like then I definitely need to take more proof based math as part of my computer science degree.
im not sure how accurate my portrayal of the proofs are but if youre intrigued i can definitely recommend it, especially with analysis topics the approach tends to give you a much deeper understanding for whats actually going on
Great video. Starting my maths journey and this is very helpful
all the best!
first time I've heard _Baba is Me_ music in an unrelated youtube video
Cool video, goat montage
Today is my maths exam. Wish me luck!
I will bring 4 gpa.
good luck dude!
@@filipus098 thanks.
Good luck
good luck friend!
I would subscribe but you are at 3.14 k subs. Love this video btw; there are not enough videos about symbolic logic on TH-cam.
Great video
👍
Wow, thanks zeropercent! This helped me prove Fermat’s Last Theroem! I would include it in this comment, but sadly it’s too long
Very interesting video! I will be taking my first analysis class (convex analysis & optimization) and was a bit worried regarding the notation. Thanks!
best of luck!
Really appreciate you using valhalla music
This would have been very helpful on my first year of uni
Very interesting, ty a lot
Tha Baba is you ost is so good
Nice Video, I think this explains. I'm a Computer Science Student, I already went through this.
Proof by Contradiction is bitch, because it's unintuitive. I hoped you would go s bit more into the detail, because I struggled a lot with this.
As far as I understand it know, we want the statement to be true. That's why we can say that if the result is wrong, the condition must also be wrong.
It's like a second hidden layer most people say nothing about. Same thing about statements without any quantifier.
Thanks for the feedback!
And also yeah for implications specifically it might be helpful to review the truth table (which i probably should also have covered in this video now that i think about it) for that operator specifically, then it should make sense why we want both the result and the condition to be wrong.
Eh its like assuming that something you want to be true isnt and then showing that leads to an absurdity.
Eg:
Not INF many primes, ie can be expressed as {p1, p2, ... pn}
What about some val k = p1*p2*p3...*pn + 1. This is clearly larger than any element in our set of primes; so it would be composite.
By fundamental theorem of arithmetic, it can be written as a product of primes, since it is not prime itself, but composite.
Take any one of those nums from that set of primes, and divide our k value by it; it should be fine. But it's not, because we are adding 1/(some prime >= 2). So our k should be able to be divided by a prime, but it isn't. This is absurd, so the set of primes we claimed actually isn't all of the primes.
(Here Euclid made k bigger than all of the primes by multiplying them, and adding the 1 to set it up so it wouldnt divide evenly, as the smallest prime is 2).
I have also seen this a lot with Well ordering principle. My god, that principle was abused the hell out of in the first half of this quarter and ik i still need to remember the proofs for the final too.
Bezout's said:
ax + by = min{ax' + by' E N: x', y' E Z}
and we could show there is a min by making x' = a, y' = b, = a^2 + b^2 which is > 0, since when doing the gcd(a, b) [which is what youre proving in bezouts], at least one of a or b is nonzero.
Then we have this ax + by and set up an inequality with gcd(a, b) one way by using the definition of gcd(a, b), and an inequality with the division algo. In the division algo part, you end up getting some "r" value from the bq+r that *would* be the new min element of the set we defined. But we already defined ax+by as the min element, so this r cannot exist in N. Hence, it = 0.
[This is just a layout of it. I liked watching learnifyable for these proofs]
Div algo proof itself used this idea also when assuming for sake of contradiction that r >= b, so r - b >= 0, but this would be min element. and for showing uniqueness.
Oh yeah or irrationals:
assume rationality, show impossible. Ie, sqrt(2) = a/b, a, b E Z, and b != 0.
Then doing some ops: 2b^2 = a^2. Well since square numbers are only div by 2 if they are even, then a is even.
2b^2 = (2k)^2, k E Z
2b^2 = 4k^2
b^2 = 2k^2.
Now we have the same thing, and can do that rule to b. But that's kinda crazy, is it not? we can keep expressing it in terms of 2*itself? We keep on dividing the numbers in question by 2, essentially, like how we said a = 2k, so k = a/2. This is crazy, at some point if we keep dividing by 2, we will not have an integer, then the equality could no longer hold true. This is also known as a disproof "By infinite descent". I know it because the name is hella cool.
TLDR: Unlikely u will come up with these early on yourself. as you learn, it is better to see examples than to struggle through with it yourself. All skills require some degree of memory.
The title is rather ambitious ?
As others pointed out you showcase proof by contrapositive instead of by contradiction
If you want to prove A=>B, this statement is equivalent to not B => not A
The negation of A=>B is "A and not B", proof by contradiction takes those and leads it to a contradiction such as x=/=x, 1=0,...
A lot of proofs for implications by contradiction can be easily turned into proofs by contrapositive since many do not use both A and not B in their deductions, only using not B and the contradiction being "since we assumed A and got not A we have a contradiction"
This only really matters for philosophical reasons about validity of the law of the excluded middle and the fact that we are assuming math is consistent, but that is the kind of nitpicky math stuff I personally love
1:45 "... [an AND expression] evaluates to true..."
This is, in my mind, where type theory/constructive logic separates from classical logic: `and` doesn't "evaluate" to anything, but it *is* **proven** when both inputs are proven.
(Note, I'm not saying you're wrong -- you're absolutely correct in the classical context --, but I'm highlighting a difference between the classical framework, and a constructive one)
Omg no way baba is you music in a math video lmao. Also, what program did you use for the animations?
blender for the 3d animations, manim for the math/latex animations and then edited everything in davinci resolve
Very popular methods lecturers use are proof by it's obvious or proof by it's a task for the student to do in home
how these visualizations are made? btw great video !
manim for everything 2d/math stuff, blender for general 3d animations and editing in davinci resolve
thanks!
@@0prcent impressive work :) waiting for more
@@0prcent Thanks man
If you are a nuroscientist watching tjis or someone you know please aske them this or let us know why we feel sleepy when watching scientific or mathematical explanations content or lectures , is it lack of sleep or how we think during the solution or paying attention or something else completly? im genuinly curious
Please continue using cat memes and Minecraft in your editting; it succeeds in making the video (and therefore, its subject) more accessible, and they're adorable!
aw thank you! glad that helped with not making it seem too overwhelming
I have the prooof for the Riemann Hypothesis but there is not enoigh space in the comment to write it.
another way of writing if and only if in a sentence instead of mathematical notation is iff. You can say A is true iff B is true
The subject of a proof is a statement that can be true, or false, or undecidable 😉
👌
Great video! However, a few nitpicks.
I think someone mentioned that the "Proof of Contradiction" you talked about is actually Proof by Contrapositive. I do think that the entire section on implications is a bit messy. For example, its confusing to call the contrapositive as the "negation of the implication", as well as the visuals seemingly implying that as well (i.e. both "Condition → ~Result" and "~Result → ~Condition")
The connection between if-and-only-if part and implications might be misleading as well. The implications you showed in the video are statements in the object logic, whereas the iff-definitions are in the metalogic.
Finally, I do think bringing up negation would be helpful, especially during the latter half. It ubiquitous everywhere, even moreso than "and" and "or".
i agree the whole thing about the names of different parts of an implivation even got myself confused while i was making the video, should have made that clearer.
i dont think the different usages of if only if would be that confusing to a beginner since it wouldnt really matter to them in the first place
and i was thinking about also including negations somewhere but it honestly felt a bit too intuitive for me to specifically talk about them, maybe thats not the case but lets see what other people think as well
thanks a lot for taking the time to write down mistakes/things to improve like these, ill take everything into consideration and i really appreciate it!
Baba is you OST at the start of the video.
Can you teach us how to proof more advanced topics in mathematics too?
There are some things i didnt cover here but as a next step it would probably be best to take a look at some actual
proofs (my recommendation would be proof-based book tackling a beginner area of mathematics, i started with analysis back then). I could do another video about advanced proof techniques/tricks but my next video will most likely be about something very different and after getting the basics of proofs down, usually the hardest part is just having an idea
how to prove anything:
1. make an assertion
2. write "QED"
you have now proved (1).
QED
I think that the Z symbol you used to denote the set of whole numbers is actually used to represent the set of integers. Integers include negative numbers where as whole numbers don't. I don't belive there is a good notation for the set of whole numbers.Generally I'll use N union {0}. Where N is the set of natural numbers.
oh just looked it up and i didnt know that "whole numbers" was an ambiguous term. integers is of course what i meant but might be a bit of a translation mistake then, thx for making me aware of this
Does formal proofs using computer theorem provers like LEAN4 ever come up in such courses?
i havent really heard of any solvers being covered at uni. not sure what kinda course you would have to attend for that but it would certainly be an interesting topic
@@0prcent Looks like there are only independent efforts by some professors th-cam.com/video/FDx0nXFQloE/w-d-xo.html and there is also the Xena project. I think it might be useful to include it in education because it looks like there is future potential for it. Imagine being a first year student playing that game adam.math.hhu.de/#/g/leanprover-community/nng4
I don't accept the law of the excluded middle. At least not without some qualifications on what objects are involved.
now i am subscribed to lowestpercent *and* zeropercent
now i will be able to prove that i cannot prove this statement!
Baba is you
How to prove anything that can be proven (in mathematics)
Your students seems like a pawns, lol. Thanks a lot.
bros a hater
@@filipus098 what that mean!
As an autodidact, this is a pretty good explanation.
Just leave it as an exercise for the reader
Step 1: Accept reality as real... Or MAYBE
Step 1: Assume that you can accept your own assumptions as acceptable. OR maybe
Step 1: Accept that concepts exist~~~~~
UUh... Maybe...
Step 1: Be alive.
Step 2: Watch the video.
its pretty funny bc in france we do this in 9th grade
0:58 0:59 1something
oh wow, set theory.
baba is you
How to prove anything:
step one: assume 1 = 2
2^3 = 8 < 9 = 3^2
proof: I made it up
cool vid btw
thx!
proof: it was revealed to me in a dream
Did you know that 1+1 = 1. ?
In bools algebra that is.
8:18 You didn't actually prove the statement is true for all n greater than or equal to 0, since when is between 2 and 4 we get false results. For example 2^2>2^2 is false; 2^3>3^2 is false; 2^4>4^2 is also false.
I don’t have to do math
This video is proof of the creators genius.
my biggest issue with proofs is no one bothers to spell out exactly what they are doing explicitly. too often relying on muh context and muh read my mind. i like programming because you have no choice but to spell out explicitly every minutia of your intended meaning. if only the math world could catch up but it is too bogged down by natural language explanations.
i like what languages like coq and agda are going.
i know exactly what you mean, i also come from a programming background and the biggest thing im missing in maths is descriptive variable names. in lots of longer proofs its impossible to track all of the A,B,C,D,x,y,z and 10 different greek letters all being used in conjunction.
Well if i learned anything it definitely proved that I'm a moron and a dumbass. Since i didn't understand a thing despite watching it 3 times
Only because the statement does not work for some negative numbers, it doesn't immediately imply that it is wrong. I don't get why this "proof" should be enough. Btw. n^2>=1 is not true for all negative numbers.
remember, n is always an integer
What would be point of math if AI or some else technology will write proofs itself like an oracle or omniscient god? It's kinda math philosophy question because all history humanity lived in some math deficit of proved knowledge even if we had many proved things we always have something to proof
…and YT decided to shadowhammer my comment
(and it seems this one got too)
edit: this one only partially
🙌 P r o m o s m
österreicher?
Hab ich auch überlegt
megga bump
First assertion is wrong. Try again.
"How to prove anything"
How do I prove consistency of ZFC?
❤🔥❤🔥❤🔥