Suppose `X` and `Y` are two sets and `f:XtoY` is a function. For a subset A of `X`, define `f(A)` to be the subset `{f(a):aepsilonA}` of `Y`. For a subset `B` of `Y`, define `f^(-1)(B)` to be the subset `{xepsilonX:f(x)epsilonB}` of `X`. Then prove the followings:
(i) Statement `"f^(-1)(f(A))=A` for every `AsubX"` is false
(ii) Statement `"f^(-1)(f(A))=A` for every `AsubX` if only if `f(X)=Y''` is false
(iii) Statement `"f(f^(-1)(B))=B` for every `BsubY"` is false
(iv) Statement `"f(f^(-1)(B))=B` for every `BsubY` if only if `f(X)=Y"` is true

Explanation

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