inverse imageInverseImage2001-10-21 01:44:372003-07-30 04:59:09Definition\usepackage{amssymb}
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\usepackage{xypic}Let $f: A \longrightarrow B$ be a function, and let $U \subset B$ be a subset. The {\em inverse image} of $U$ is the set $f^{-1}(U) \subset A$ consisting of all elements $a \in A$ such that $f(a) \in U$.
The inverse image commutes with all set operations: For any collection $\{U_i\}_{i \in I}$ of subsets of $B$, we have the following identities for
\begin{enumerate}
\item Unions:
$$f^{-1}\left(\bigcup_{i \in I} U_i\right) = \bigcup_{i \in I} f^{-1}(U_i)$$
\item Intersections:
$$f^{-1}\left(\bigcap_{i \in I} U_i\right) = \bigcap_{i \in I} f^{-1}(U_i)$$
\end{enumerate}
and for any subsets $U$ and $V$ of $B$, we have identities for
\begin{enumerate}
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\item Complements:
$$\left(f^{-1}(U)\right)^\complement = f^{-1}(U^\complement)$$
\item Set differences:
$$f^{-1}(U \setminus V) = f^{-1}(U) \setminus f^{-1}(V)$$
\item Symmetric differences:
$$f^{-1}(U \bigtriangleup V) = f^{-1}(U) \bigtriangleup f^{-1}(V)$$
\end{enumerate}
In addition, for $X \subset A$ and $Y \subset B$, the inverse image satisfies the miscellaneous identities
\begin{enumerate}
\setcounter{enumi}{5}
\item $(f|_X)^{-1}(Y)=X\cap f^{-1}(Y)$
\item $f\left(f^{-1}(Y)\right) = Y\cap f(A)$
\item $X \subset f^{-1}(f(X))$, with equality if $f$ is injective.
\end{enumerate}