# Proof: Inverse Functions

• Sep 1st 2012, 08:34 PM
lovesmath
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!
• Sep 2nd 2012, 02:46 AM
Plato
Re: Proof: Inverse Functions
Quote:

Originally Posted by lovesmath
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 \$\displaystyle X=\{1,2,3,4,5\},~A=\{4,5\},~Y=\{a,b,c,d\},~B=\{a,b \}~\&\$
\$\displaystyle f=\{(1,a),(2,b),(3,b),(4,c),(5,d)\}\$ but \$\displaystyle f^{-1}(f(A))=A,~\&~f(f^{-1}(B))=B.\$
Please review the statement for errors.