Proofs by contradiction.

[@goodplacetostop2973]

0:34 Good Place To Start 17:03 Good Place To Stop

[@damianbla4469]

00:34 "This is a good place to... start"? Nobody expected that. I am shocked :O

[@SubtleForces]

I guess it had to be expected in a video about proof by contradiction

[@orenfivel6247]

00:34 "This is a good place to... 🌟"

[@ziquaftynny9285]

I don't get it

[@DavidCorneth]

7:21 a video by Michael Penn showing with overkill there are infinitely many primes: kzbin.info/www/bejne/i4a8f5KIhrJsj80

[@muimerp3]

I think one good way of teaching reasoning by contradiction is with Sudoku. Sometimes, if you suppose that in some square a given number should exist, you arrive at the conclusion that, say, in one of the rows of the game such number cannot exist. Obviously, a contradiction. Therefore, that number cannot exist in the square where you supposed it would exist.

[@RandomBurfness]

If I'm writing a proof and in the middle of a proof, I argue by contradiction I like to enclose that argument in a little box and then end the argument with a little lightning bolt symbol striking down for the fun of it all.

[@punditgi]

Michael, what textbook are you using?

[@sapientum8]

Excellent content. Need more videos on mathematical thinking, because the vast majority of the viewers have not yet reached mathematical maturity.

[@godfreypigott]

Here is a copy-paste of my comparison from two days ago of proof by contradiction and proof by contraposition. This applies when the result to be proven is not a conditional - it would need a slight modification if the result is a conditional *:* Contradiction and contraposition are logically the same. The difference is only in the details. In a proof by contraposition, the premise P is stated in, or implied by, the question. In a proof by contradiction, no P is given to you - it's up to the proof-writer to come up with that themselves. In the proof that sqrt(2) is irrational, this as yet unproven fact is Q, and P is not yet known. You then assume ~Q, and try to prove ~P, where P is still not known. But eventually in your proof you get to get to a statement which is obviously false (such as 1=2), and when you do you call that statement ~P. So you have proven ~Q => ~P, which by contraposition is the same as P => Q. But P is tautologically true, so you have proven Q.

[@mathflipped]

Many people don't realize that the proof by contrapositive is the same as the proof by contradiction but phrased slightly differently.

[@noahtaul]

Tiny comment: proofs by contrapositive of P->Q have you proving ~Q->~P, so in your proof you only use ~Q, never P. Proof by contradiction you can use both ~Q and P and work to try to find a contradiction. Contrapositive is like a proof by contradiction where the ending is “... so ~P. But we assumed P, contradiction, so Q is true.” and within the proof you never actually use P.

[@linggamusroji227]

It's already different from the definition

[@mathflipped]

@@noahtaul And this is why proof by contrapositive is redundant. It can always be replaced with a proof by contradiction.

[@noahtaul]

@@mathflipped I agree! Direct proofs are the same way. Although sometimes there’s something aesthetically pleasing about doing writing in non-contradiction ways- you end the proof with the statement you want to prove, and the whole proof is just moving forward one step at a time, never making assumptions to be invalidated later

[@cobalt3142]

The way I've learned it/my convention: to prove P->Q, proof by contrapositive requires you to assume ~Q, then show that this implies ~P. Proof by contradiction, on the other hand, would have you assume both P and ~Q, and then for some other statement R show that this implies both R and ~R. I've always held this distinction, but I guess it's also fair to say that contrapositive is the same as contradiction where you show that the original statement would be the contradiction

[@MathamaniaMata]

I like this version of the infinite primes proof. I always wondered why we used that number until I realized that we’re always going to be adding new primes to the list, thus creating our contradiction. Neat! Thanks, Michael!

[@Carmenifold]

love that "1/2 is an integer" contradiction, that was fun

[@schweinmachtbree1013]

thought I'd point out that Michael said that there was already a contradiction because the LHS was even and the RHS was odd, but proving the fact "even =/= odd" (which is so "obvious" that one often doesn't even think to prove it) amounts exactly to the contradiction that 1/2 is an integer: if even = odd them 2n = 2m+1 for integers n and m, hence 2(n-m) = 1, hence 1/2 = n-m, an integer :)

[@trelligan42]

REQUEST: May we please have a playlist of these "proof" videos? I'm sure I'm missing one...

[@goodplacetostop2973]

kzbin.info/aero/PL22w63XsKjqykuLOimt7N59e6wn_aE0wd

[@trelligan42]

@@goodplacetostop2973 Thank you very much.

[@dipeshpatil2771]

17:03

[@samandriod5318]

Keep up the good work.

[@punditgi]

So, if you want to be truly international, Michael, you should start saying, "zed" instead of "zee"!

[@alejandrolagunes5697]

Why's that?

[@punditgi]

@@alejandrolagunes5697 Because every English speaking country says "zed" ... except one. Care to guess which one? Kind of makes us an outlier. Same story when it comes to using metric, or more precisely not.

[@alejandrolagunes5697]

@@punditgi i didn't know that, i was taught letter z goes by "zee" in school, as I'm from a spanish-speaking country

[@punditgi]

@@alejandrolagunes5697 Sure. Evidence of American influence. If you had been taught English by a Canadian, a Brit, an Irishman, an Australian, a New Zealander, a South African, etc. you would have learned "zed".

[@anshumanagrawal346]

Lol, right even in India we say Zed and not Zee