Paper 4, Section II, E

Numbers and Sets | Part IA, 2009

(a) Let AA and BB be non-empty sets and let f:ABf: A \rightarrow B.

Prove that ff is an injection if and only if ff has a left inverse.

Prove that ff is a surjection if and only if ff has a right inverse.

(b) Let A,BA, B and CC be sets and let f:BAf: B \rightarrow A and g:BCg: B \rightarrow C be functions. Suppose that ff is a surjection. Prove that there is a function h:ACh: A \rightarrow C such that for every aAa \in A there exists bBb \in B with f(b)=af(b)=a and g(b)=h(a)g(b)=h(a).

Prove that hh is unique if and only if g(b)=g(b)g(b)=g\left(b^{\prime}\right) whenever f(b)=f(b)f(b)=f\left(b^{\prime}\right).

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