Subsets and Relations

Question

Prove the following:

Let f: X->Y be a mapping from X into Y.

Show that if A and B are subsets of X, then

$(A⊂B)⇒(f(A)⊂f(B))$

but $(f(A)⊂f(B))$ does not imply that $(A⊂B)$

and if A' and B' are subsets of Y, then

$(A′⊂B′)⇒(f^{-1}(A′)⊂f^{−1}(B′))$

Thank you for any help!

edit: they're from Mathematical Analysis by Zorich

edit: sorry for the sloppiness on the initial post.

Answer 1

For the first one we have: $A\subset B$. Take $x\in f(A) \Rightarrow f^{-1}(x)\subset A \subset B$.

So $x \in f(B)$.

For a counter argument use the function: $f(0)=0,f(1)=0,f(2)=1$ and $A={0},B={1,2}$

Then: $f(A)=\{0\}\subset \{0,1\}=f(B)$ but $A=\{0\}\not\subset \{1,2\}=B$

For the second one we have:

$A'\subset B'$

Take $x\in f^{-1}(A') \Rightarrow f(x)\in A' \subset B'$.

So: $x\in f^{-1}(B')$