Proof By Cases | 2 Examples

Two examples of using the proof by cases method.
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[@NikolajKuntner]

Very nice. Here's my somewhat "ring theoretic" take on the first problem: In the natural numbers, every number g generates an ideal, namely the g-spaced grid G={...,-2·g, -g, 0, g, 2·g, 3·g, 4·g, ...}. For example, the number 4 generates the 4-spaced grid {...,-8, -4, 0, 4, 8, 12, 16,...}. An ideal is defined by the fact that multiplying an element of the ideal by any number, you get back into the ideal. E.g. 8 is in the ideal and we see that 33·8 is also in the 4-spaced ideal, namely it's the 66'th element of the grid after 0. To say a number is "even" means it's a multiple of 2, which in turn means it's in the 2-spaced ideal. Indeed, for g=2, multiplying any even number by any number gives again an even number. So let a=3 and b=5 be the coefficients in our sum n^2 + a·n + b. Both a and b are odd. To show that this sum is always odd is the same as showing that n^2 + a·n is always even. We can rewrite this sum as n·(n + a). If n is even, we know that this expression is in the ideal and we're done. On the other hand, if n is odd, then n + a is even and we're also done by the same reasoning.

[@BrainGainzOfficial]

Interesting. I'm not taking graduate level abstract algebra until the fall. We didn't go that deep into ring theory in the undergraduate course, but what you're talking about does make sense. It sounds similar to the idea of cosets and quotient groups.

[@NikolajKuntner]

@@BrainGainzOfficial Indeed, you have a correspondence between the ideals and the equivalence relations that preserve the ring structure.

[@stopthecap8810]

subscribed and liked -- such a great video. why did you not include 0 in your first proof? could you make another video on proof by cases (specifically uniqueness/existence proofs)?

[@larsson4488]

You are really good at teaching!

[@AshishSingh-753]

Thanks Sir for explaination

[@user-db3zv2de4q]

Nice

[@mysteryman07019]

What chalkboard do you have and what kind of chalk do you have!

[@BrainGainzOfficial]

I use Hagoromo chalk! I don't remember the brand of the chalkboard, but I know I got it off amazon.

[@mysteryman07019]

Brain Gainz oh sick! I appreciate that, thank you!

[@arjay_2002]

what do you mean when you say "by definition"?

[@user-db3zv2de4q]

Pfff “ mind blowing, right “😂. That’s how teaching should be

[@NaomiManda-um8rj]

What are the technique for proofing