# Question about group theory, cardinality and isomorphic

I would like to know how to identify and prove the cardinality of sets and how to identify isomorphic.

(See attached file for full problem description)

Group Theory:

a. If S and T are sets then let TS denote the set of all functions from S to T. Prove that the cardinality of TSxU equals the cardinality of (TS)U

b. Consider the groups Z3 x Z3 and Z9. These are each "integer groups" of order 9. Are they isomorphic or not? Give an explicit reason.

Z- integer

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#### Solution Preview

To prove that two sets A & B have the same cardinality, we prove that there is a bijection (a one-to-one and onto map) from A to B.

Here we have A = T^(SxU) = The set of functions f:(SxU)->T

B = (T^S)^U = the set of functions g:U->{the set of functions from S to T}

Let f be an element of A. ...

#### Solution Summary

This solution is comprised of a detailed explanation to prove that the cardinality of TSxU equals the cardinality of (TS)U.