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?

Thank you!