Let $f: A \rightarrow B$ be a function, and $C \subset A$. Define the **image** (or **direct image**) of $C$ as
$
f(C):=\{f(x) \in B: x \in C\} .
$
Let $D \subset B$. Define the **inverse image** of $D$ as
$
f^{-1}(D):=\{x \in A: f(x) \in D\} .
$
---
Let $f: A \rightarrow B$. Let $C, D$ be subsets of $B$. Then
$
\begin{array}{l}
f^{-1}(C \cup D)=f^{-1}(C) \cup f^{-1}(D), \\
f^{-1}(C \cap D)=f^{-1}(C) \cap f^{-1}(D) \\
f^{-1}\left(C^{c}\right)=\left(f^{-1}(C)\right)^{c}
\end{array}
$
---
Let $f: A \rightarrow B .$ Let $C, D$ be subsets of $A$. Then
$
\begin{array}{l}
f(C \cup D)=f(C) \cup f(D), \\
f(C \cap D) \subset f(C) \cap f(D)
\end{array}
$