Prove that $B\not\subset f(f^{-1}(B))$

Question

For a function $f:X\rightarrow Y$, where $B\subset Y$, I need to prove that $B\not\subset f(f^{-1}(B))$, but I'm not necessarily sure how to go about it. It's easy to show that $f(f^{-1}(B))\subset B$, but that's not too useful here.

Answer 1

Juse take a non-surjective function $f$ from $X$ into $Y$ and a set $B\subset Y$ such that $f(X)\varsubsetneq B$. Then, since $f\bigl(f^{-1}(B)\bigr)\subset f(X)$, $B\varsubsetneq f\bigl(f^{-1}(B)\bigr)$.

Answer 2

Let $X={\text {{1,2}}}$ and $ Y={\text {{1,2,3}}}$ and $B={\text {{1,2}}}$.

Define $f:X\to Y$ as$$f(1)=f(2)=1.$$

Note that $$f^{1} (B) ={\text {{1,2}}}$$

$$f(f^{-1} (B)) = {\text {{ 1}} } $$

Thus $$B\not\subset f(f^{-1}(B))$$