inverse image
Let $f:A\u27f6B$ be a function, and let $U\subset B$ be a subset. The 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

1.
Unions:
$${f}^{1}\left(\bigcup _{i\in I}{U}_{i}\right)=\bigcup _{i\in I}{f}^{1}({U}_{i})$$ 
2.
Intersections^{}:
$${f}^{1}\left(\bigcap _{i\in I}{U}_{i}\right)=\bigcap _{i\in I}{f}^{1}({U}_{i})$$
and for any subsets $U$ and $V$ of $B$, we have identities for

3.
Complements^{}:
$${\left({f}^{1}(U)\right)}^{\mathrm{\complement}}={f}^{1}({U}^{\mathrm{\complement}})$$ 
4.
Set differences^{}:
$${f}^{1}(U\setminus V)={f}^{1}(U)\setminus {f}^{1}(V)$$ 
5.
$${f}^{1}(U\u25b3V)={f}^{1}(U)\u25b3{f}^{1}(V)$$
In addition, for $X\subset A$ and $Y\subset B$, the inverse image satisfies the miscellaneous identities

6.
${({f}_{X})}^{1}(Y)=X\cap {f}^{1}(Y)$

7.
$f\left({f}^{1}(Y)\right)=Y\cap f(A)$

8.
$X\subset {f}^{1}(f(X))$, with equality if $f$ is injective^{}.
Title  inverse image 
Canonical name  InverseImage 
Date of creation  20130322 11:51:58 
Last modified on  20130322 11:51:58 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  10 
Author  djao (24) 
Entry type  Definition 
Classification  msc 03E20 
Classification  msc 46L05 
Classification  msc 8200 
Classification  msc 8300 
Classification  msc 8100 
Synonym  preimage 
Related topic  Mapping 
Related topic  DirectImage 