This week we reviewed our proof techniques by showing some interesting things about the triangle inequality, sets, and DeMorgan’s law.
The triangle inequality states that for real numbers
|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.]
x + y = |x + y| ≤ |x| + |y| = x + y, which you can see is true because
x + y ≤ x + y.
|x + y| ≤ |x| + |y|because
|x + y| ≤ x.
-x - y = |x + y| ≤ |x| + |y| = -x - ybecause
-x - y = -x - y.
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.
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.
This one law applies to both sets and logic? If you look at the two examples, those seem similar. Replace
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!