I need to the prove the following:
Prove that if A and B are finite sets, then A ≈ B if and only if |A| = |B|.
This is confusing me because doesn't the definition of numerically equivalent actually say that the cardinality of A = the cardinality of B? If so, does that mean I can just say "By the definition of numerically equivalent"?
Nope, A ≈ B iff there exists a function θ that is 1-1 and onto.
It's actually a lot easier than it seems. What you need to do is define the sets A and B without a loss of generality. So just make up names for them. a0, a1, a2, ..., a_n are all in A and b0, b1, b2, ..., b_n are all in B. You know they each go to n since they are finite and have the same number. Then let θ:A->B such that θ(a_k) = b_k. It should be really easy to show 1-1 and onto with this map.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."