10.5, Due November 14

Difficult: So, Schroder and Bernstein proved their theorem without Theorem A, and therefore without the axiom of choice? Perhaps could you expand exactly what the axiom of choice is and means. Is it required since Schroder and Bernstein proved their theorem without it?

I actually understood the two functions in theorem 10.19 , as in how they go from numbers in (0,1) --> P(N) injectively and P(N)--->(0,1) injectively. However, I doubt I would ever be able to prove it as they did. However, it does make sense intuitively what they did.

Interesting: I love how the Scroder Bernstein theorem uses concepts and theorems from throughout the whole chapter and previously!! It comes together so nicely!

It's interesting how we use the interval (0,1) for most of the proofs for the cardinailty of the reals. But it always work out, since we proved there is a bijective function between (0,1) to the reals, so whatever is equal to the cardinality of (0,1) is also equal to the cardinality of the reals!