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'inverse image'
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| Title of object: |
inverse image |
| Canonical Name: |
InverseImage |
| Type: |
Definition |
| Created on: |
2001-10-21 01:44:37 |
| Modified on: |
2003-07-29 16:22:00 |
| Classification: |
msc:03E20 |
| Synonyms: |
inverse image=preimage |
Revision comment (for changes between this and next version):
| Changes for correction #2290 ('notational mixup'). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic} |
Content:
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 $Y$, we have identities for
\begin{enumerate}
\setcounter{enumi}{2}
\item Complements:
$$\left(f^{-1}(A)\right)^\complement = f^{-1}(A^\complement)$$
\item Set differences:
$$f^{-1}(A \setminus B) = f^{-1}(A) \setminus f^{-1}(B)$$
\item Symmetric differences:
$$f^{-1}(A \bigtriangleup B) = f^{-1}(A) \bigtriangleup f^{-1}(B)$$
\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 $(f|_A)^{-1}(B)=A\cap f^{-1}(B)$
\item $f\left(f^{-1}(B)\right) = B\cap f(X)$
\item $A \subset f^{-1}(f(A))$, with equality if $f$ is injective.
\end{enumerate} |
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