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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Abstract algebra proof: composition of mappings

**Physics Forums | Science Articles, Homework Help, Discussion**