I have two problems:
1. Let A, B, and C be sets. Suppose that card A < card B and card B < card C. Prove that card A < card C.
Can I do this by contradiction?:
Assume card A < card B, card B < card C, and by contradiction, that card A > C.
We are give that card A < card B.
By transitivity of the relation <, card B < card C and card C < card A implies that card B < card A.
Contradiction, because card A < card B and card B < card A can't both be true.
Is this right or am I missing something?
2. Let K be any set, and let F be the set of all functions with domain K. Then card K < card F.
I don't know how to start with this one. It makes sense, because there are infinitely many functions for each possibility of the codomain... But I don't know how to say that. Any hints?