Numbers and Sets | Part IA, 2008

Let A,BA, B and CC be non-empty sets and let f:ABf: A \rightarrow B and g:BCg: B \rightarrow C be two functions. For each of the following statements, give either a brief justification or a counterexample.

(i) If ff is an injection and gg is a surjection, then gfg \circ f is a surjection.

(ii) If ff is an injection and gg is an injection, then there exists a function h:CAh: C \rightarrow A such that hgfh \circ g \circ f is equal to the identity function on AA.

(iii) If XX and YY are subsets of AA then f(XY)=f(X)f(Y)f(X \cap Y)=f(X) \cap f(Y).

(iv) If ZZ and WW are subsets of BB then f1(ZW)=f1(Z)f1(W)f^{-1}(Z \cap W)=f^{-1}(Z) \cap f^{-1}(W).

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