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Math Help - Proof: Inverse Functions

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    Proof: Inverse Functions

    Let X and Y be sets and f:X->Y be a function. For A is a proper subset of X and B is a proper subset of Y, recall the definitions of f(A) is a proper subset of Y and f^-1(B) is a proper subset of X,
    f(A)={f(a):a is an element of A} and f^-1(B)={x is an element of X:f(x) is an element of B}

    Prove:
    A is a proper subset of f^-1(f(A)) and f(f^-1(B)) is a proper subset of B; equality need not hold in either case.

    Not even sure where to begin. Help, please!
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    Re: Proof: Inverse Functions

    Quote Originally Posted by lovesmath View Post
    Let X and Y be sets and f:X->Y be a function. For A is a proper subset of X and B is a proper subset of Y, recall the definitions of f(A) is a proper subset of Y and f^-1(B) is a proper subset of X,
    f(A)={f(a):a is an element of A} and f^-1(B)={x is an element of X:f(x) is an element of B
    Prove:
    A is a proper subset of f^-1(f(A)) and f(f^-1(B)) is a proper subset of B; equality need not hold in either case.
    Not even sure where to begin. Help, please!
    This is not true.
    Let X=\{1,2,3,4,5\},~A=\{4,5\},~Y=\{a,b,c,d\},~B=\{a,b  \}~\&
    f=\{(1,a),(2,b),(3,b),(4,c),(5,d)\} but f^{-1}(f(A))=A,~\&~f(f^{-1}(B))=B.
    Please review the statement for errors.
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