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    Question about group theory, cardinality and isomorphic

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    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.