define right-inverse of a mapping B to be mapping A, such that B * A= identity (iota). Where the operation * is composition. Note that B is A's left-inverse.(adsbygoogle = window.adsbygoogle || []).push({});

QUESTION:

Assume S is a nonempty set and that A is an element of M(S) -the set of all mappings S->S.

a) Prove A has a left inverse relative to * iff A is one-one

b) Prove that A has a right inverse relative to * iff A is onto.

ANSWER:

I answered a) to the best of my ability, using firstly a theorem that states (B*A is 1-1) -> A is 1-1. Then, I simply constructed B from A (since A is 1-1) to prove the converse.

b) on the other hand, i found a little harder. Once again, i used a theorem that said A*B is onto -> A is onto. Now, though, I can't seem to prove the converse. The question is, how can i construct B, knowing only that A is onto? any help would be greatly appreciated.

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# Abstract algebra proof: composition of mappings

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