Proof portfolio II questions. The first two lemmas provide an alternate way to prove that two functions are inverses of each other. If A is a set, define the identity map on A by idA: A→ A, idA (a) = a. Then idA is clearly a bijection. Lemma 1 Suppose that f: A→ B and g: B→ A satisfy go f = idA, and fog = idB. Then f and g are bijections. Hint: By symmetry you can just prove that f is a bijection. Do this directly from the definitions (don't quote any other results).

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