Question

4. Let X and Y be any sets and let F be any one-to-one (injective) function from X to Y. Prove that for every subset A CX: (a) (10 points) AC F-(F(A)). (b) (10 points) F-1(F(A)) C A.

Answer 1

Note that for any subset A of X, F(X) = { F(x) $\in$ Y : x $\in$ A } and that,
for any subset B of Y, F^-1(B) = { a $\in$ X : F(a) $\in$ B } is the inverse image of B under F.

(a)
x $\in$ A $\Rightarrow$ F(x) $\in$ F(A) $\Rightarrow$ x $\in$ F^-1(F(A))
This shows that A $\subset$ F^-1(F(A)).

Note that the above relation always holds for any function F, let alone an injection.


(b)
This part uses the injectivity of F.
Suppose x $\in$ F^-1(F(A)). Then, y = F(x) $\in$ F(A). By definition of F(A), there exists a z $\in$ A such that F(z) = y = F(x).
By the one-to-oneness of F, it follows that x = z. Thus, x $\in$ A.
Since x $\in$ A is arbitrary, hence F^-1(F(A)) $\subset$ A.


Thus, by the Axiom of Extension (Two sets are equal if and only if they have the same elements),
F^-1(F(A)) = A.