10.5 Part One, Due November 12

Difficult: Holy cow. Theorem 10.17 is confusing me for sure!! I'm trying really hard to think of an example to run this through with, but I cannot think of one... Let me write out one..

A: {Integers}
B:{Evens, 1, 5}
f:A-->B such that f(x)=2x
ran f is a subset of B, specifically ran f= {Evens}
A-B={Odds, except 1 and 5}
B'={Evens, except 2 and 4 and 8 and 16 and 32... and except 10 and 20 and 40 and 80...}
C=Union of two sets above
D={1,5, and 2,4,8,16,32... and 10,20,40,80...}

So h(x)= {f(x) if x in C, and i_D(x) if x in D

Woah. I think that worked. That is a bijective function from A to B. However, that is a lot of sets to consider and EXTREMELY complicated. There has to be an easier way to think about this... Maybe you can show a less complicated example? (Or worse, they're always complicated...)

Interesting: Well, I find it very cool that I was able to conjure up an example of Theorem 10.17. The Schroder-Bernstein Theorem is also extremely interesting, because it utilizes concepts from four different theorems to come up with an intuitively conceivable fact.