Now Prove It!

This week we reviewed our proof techniques by showing some interesting things about the triangle inequality, sets, and DeMorgan’s law.

The Triangle Inequality

The triangle inequality states that for real numbers x and y, |x + y| ≤ |x| + |y|. We can prove this by looking at a few cases: footnote:[I’m not going to structure it like a formal proof, because, quite frankly, I’m tired of them and I don’t think it helps you to understand what’s going on.]

Why is it called the triangle inequality? Because this can also be shows using triangles. |x| is the length of one side, |y| of the other, and |x + y| is the length of the hypotenuse. Clearly, if all of the edges of the triangle are to connect, this equality must hold true. footnote:[Yes, this means that the triangle inequality was derived using vector math, but since isn’t well defined for vectors, it’s expressed in terms of their lengths.] This doesn’t seem super exciting, but it turns out to show up a lot when proving other things, so I wanted to mention it so that the tool would be in your head, to mix a metaphor.

DeMorgan’s Law

The other really amazing part of this week was actually something my professor just mentioned offhand. First, some background. If you have a logical statement (remember those) not (P and Q), that’s equivalent to not P or not Q. That’s DeMorgan’s law, as it applies to logic. Also remember that sets have a complement, that is, the set of all things not in the first set. Okay. Now comes the mind-shaking part that my professor wrote on the board. The complement of (A intersection B) = The complement of A union the complement of B. footnote:[He used actual symbols, of course, but I heard from some people that the union and intersection symbols weren’t showing up for them. I’m working on it.] Then, the professor called that DeMorgan’s law.

Wow.

This one law applies to both sets and logic? If you look at the two examples, those seem similar. Replace not with complement, or with union and and with intersection. Does this reveal a deeper symmetry’s between logic and set theory? Maybe! I’m not yet sufficiently knowledgeable, so I don’t know why this similarity occurs or how to prove it, although I have some ideas involving looking at membership of statements in sets …

Regardless, it was an exiting realization. I hope you join me next week for another bout of math!