Proof by Contrapositive: If n^2 is Even then n is Even
ฝัง
- เผยแพร่เมื่อ 28 ส.ค. 2024
- Using the contrapositive, we prove that if n^2 is even then n is even. A proof by contrapositive is not necessary here, we'll touch on how it could be done directly, but this is definitely a case where the contrapositive is more obvious. We'll also recap what the contrapositive actually is, and why we care about it. #Proofs
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you are single handedly getting me through my discrete mathematics class lol. thank you, super helpful content
Happy to help! good luck!
Can't wait for the calculator documentary. Wait its out im going to watch it.
It's the journey of a lifetime!
Was able to follow along, even able to finish it!
Awesome! Thanks for watching!
such a nice proof thanks so much!
Beautifully explained 😊
Thanks a lot 😊
Can you explain if n² is even, then a is even
The fact that 2 is already in n (it’s a factor of n) and, therefore, makes n an even number was hard for me to grasp the first time around.
Yeah that's something that can be hard to grasp at first, but it's one of those things - once you see it, it makes perfect sense! The 2 can't spontaneously generate, so it was there all along. That fundamental theorem of arithmetic is a powerful thing.
Can you make a video on proving the fundamental theorem of arithmetic
Thank you so much ❤
No problem!
Could you proof this same statement using the direct method?
Nice explanation nice proof
Thanks .
Thanks for watching!
*provided n is a integer
Haha, I should have just included that in the original statement! Hopefully it won't be confusing for people!
In this case couldn't you just prove the converse, i.e. "when n is even then n^2 is even" ?
Thanks for watching and good question! We could do that, but it wouldn't help us in proving our original result. Remember - the converse of a statement is different from the original statement. So if we proved n being even forces n^2 to be even, that'd be fine, but it wouldn't tell us much about what is true if n^2 is even. We know n COULD be even if n^2 even, but at that point we still wouldn't know if n could be odd or not.
Here is a non-math example. If it is raining then the ground is wet. This does not mean that if the ground is wet then it is raining, since a sprinkler could be on, or we could have just had a crazy water balloon fight, or the neighbor's dog could have relieved himself on the ground.
@@WrathofMath Ah, thank you for the explanation! I will start my first semester at university studying math in September so I don't have so much experience with logic yet.
That's awesome, I hope you like it! Two very important things you'll learn in logic are 1) the converse is different, and 2) the contrapositive is the same. We talk about statements like "If P then Q", whose converse is "If Q then P" and whose contrapositive is "If not Q then not P".
I have some videos on logic, but will certainly do more at some point. I don't know what text you'll be using, but recommendations from me are Book of Proof by Hammack (which you can get in PDF for free from his website) and Proofs by Jay Cummings (which came out this year, is very affordable and funny, and Jay Cummings appeared in my recent calculator documentary so that fills me with joy).
@@WrathofMath @SeeTv Also important, proof by contradiction is not the same as proof by contrapositive. It's a subtle distinction, but I've made that mistake more than a few times :D
Impressive
Thanks for watching!
By the way, would proof by contradiction would be sufficient in this case? It seems as it could be solved faster in such way.@@WrathofMath
Nice proof why don't you teach on youtube
I'm not sure what you mean - that's what my whole channel is for!
kk