# Let A and B be sets and let f : A &#x2192; B be a function. Show that if D

Let A and B be sets and let $f:A\to B$ be a function. Show that if $D\subseteq B$, then $f\left(f-1\left(D\right)\right)\subseteq D$.
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Step 1
Some things that may help. Say we have a function $F:A\to B$ and that $b\in B$ (b is an element of B). Then, ${f}^{-1}\left(b\right)$ is a set (yes, a set, I know that can be confusing).
The elements of the set ${f}^{-1}\left(b\right)$ are the elements $a\in A$ such that $f\left(a\right)=b$.
If you know set builder notation, then we say that ${f}^{-1}\left(b\right)=\left\{a\in A:f\left(a\right)=b\right\}$.
Now, to generalize this, let's say instead of a point in B, we have a subset of B, call it D. Then, ${f}^{-1}\left(D\right)=\left\{a\in A:f\left(a\right)\in D\right\}$.
Step 2
And, similarly, if $C\subset A$, then f(C) is also a set (this time it's f that's used to make a set, not ${f}^{-1}$). Precisely, $f\left(C\right)=\left\{f\left(c\right):c\in C\right\}$.
So, if $x\in f\left({f}^{-1}\left(D\right)\right)$, we replace C with ${f}^{-1}\left(D\right)$
$f\left({f}^{-1}\left(D\right)\right)=\left\{f\left(c\right):c\in {f}^{-1}\left(D\right)\right\}$
This should help you get started. Let me know if you have more questions.
###### Did you like this example?
mistergoneo7
Step 1
This should help: For every function $f:A\to B$ (whether bijective, injective, surjective, or none of these), we can defined a function $\begin{array}{rl}{f}^{-1}:\mathcal{P}\left(B\right)& \to \mathcal{P}\left(A\right)\\ Y& ↦\left\{\phantom{\rule{thinmathspace}{0ex}}x\in A\mid f\left(x\right)\in Y\phantom{\rule{thinmathspace}{0ex}}\right\}\end{array}$
Step 2
But also $\begin{array}{rl}f:\mathcal{P}\left(A\right)& \to \mathcal{P}\left(B\right)\\ X& ↦\left\{\phantom{\rule{thinmathspace}{0ex}}f\left(x\right)\mid x\in X\phantom{\rule{thinmathspace}{0ex}}\right\}.\end{array}$