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I am trying to prove the following about equipotence:

Let A and B be nonempty sets. We say that A is equipotent with B if there is a bijection between A and B. Then the following hold:

(i) A is equipotent with itself.

(ii) If A is equipotent with B, then B is equipotent with A.

(iii) If A is equipotent with B, and B is equipotent with C, then A is equipotent with C.

Proof:

(i) We can use the identity function Id_A which gives a bijection between A and itself. Shall I need a more formal proof here?

(ii) Let f: A -> B be a 1-1 and onto map. We can use the inverse function f^-1 which will give a bijection between B and A. Same question here, how to give formal proof.

(iii) Let f:A->B, g:B->C be 1-1 and onto. Then the composition h=(g o f) will give a 1-1 and onto map from A onto C. How do you give a formal proof of this?

Hope you guys have some suggestions.

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# Equipotence and non-empty sets

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