Two examples of using the proof by cases method. Leave any questions / comments below! Keep flexin' those brain muscles! Facebook: / braingainzofficial Instagram: / braingainzofficial
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.
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.
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)?
Hello sir! May I ask if proof by cases is the same as the choose method? I’m confused about the two. The same goes for the Construction method and direct proofs. Are those two the same as well? Please enlighten me. Thanks.
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.
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.
@@BrainGainzOfficial Indeed, you have a correspondence between the ideals and the equivalence relations that preserve the ring structure.
You are really good at teaching!
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)?
0 is an even number
Hello sir! May I ask if proof by cases is the same as the choose method? I’m confused about the two. The same goes for the Construction method and direct proofs. Are those two the same as well? Please enlighten me. Thanks.
what do you mean when you say "by definition"?
Thanks Sir for explaination
What chalkboard do you have and what kind of chalk do you have!
I use Hagoromo chalk! I don't remember the brand of the chalkboard, but I know I got it off amazon.
Brain Gainz oh sick! I appreciate that, thank you!
Nice
What are the technique for proofing
Pfff “ mind blowing, right “😂. That’s how teaching should be
Please mention the problems you saud we we will see. I don't understand 🤓🫣