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Homework Statement
Prove:
If f is injective (1-1), then f^-1 [ f(C) ] = C
Homework Equations
f: A -> B
C is a subset of A, and D is a subset of B
note f^-1(D) is the preimage of D in set A
and f(C) is image of C in set B
The Attempt at a Solution
My attempt:
Assume f is injective, WMST f^-1 [ f(C) ] = C
to show set equality, we have to show that the left side is a subset of the right side, and vice versa
I have already shown that C is a subset of f^-1 [ f(C) ]
my problem is showing that f^-1 [ f(C) ] is a subset of C
I assumed x is an element of f^-1 [ f(C) ]
by definition of preimage f^-1(D) = { x element of A | f(x) element of D}
hence, f(x) is an element of f(C)
and by definition of image f(C) = { f(x) | x element of C}
can i conclude that, since f(x) is an element of f(C), hence x is an element of C
therefore the proof is complete
but i believe this is wrong, since i have not used the fact that f is injective (1-1)
can anyone provide any hints?