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Given a function f:X→Y , where X and Y are two non-empty sets, also f(C)={f(x): x∈C} and f−1(D)={x:f(x)∈D , for some C⊆X, B⊇Y , which of the following are true?

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a

f−1(f(A))=A

b

f−1(f(B))=B only if B⊆f(X)

c

f−1(f(A))=A only if f(X)=Y

d

f(f−1(B))=B

answer is B.

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Detailed Solution

f({x,y})={a,b} but f−1{a,b} may not be {x,y}∴ f−1(f(A))≠A ∴ (A) is not trueIf B⊂f(X) =range f , then f(f−1(B))=B(C) and (d) are also not always true.

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Given a function f:X→Y , where X and Y are two non-empty sets, also f(C)={f(x): x∈C} and f−1(D)={x:f(x)∈D , for some C⊆X, B⊇Y , which of the following are true?