# Set Proof

**Functions-Sets Proof help?**

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*Let A and B be finite non-empty sets . Denote A^B as the set of all functions from B into A. Prove that the cardinality of A^B is equal to the cardinality of A raised to the power cardinality of B. *

I know this formula is correct and can be used to find the number of functions from one set to another but how do I prove this? If someone could explain this or even give me an idea I'd be very appreciative. Thanks.

I would do a recurrence on card(B).

If card(B) = n and card(A) = m.

C = B U {e} a new element.

For every function from B into A, you can easily extend this function in m different ways to a function from C into A. You have n^m different functions from B into A, you can then create m*n^m functions from C into A. I hope it's understandable.

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