inverse image

Let $f:A⟶B$ 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

1. Complements:

   $${\left(f^{-1}(U)\right)}^{\mathrm{\complement }}=f^{-1}(U^{\mathrm{\complement }})$$

2. Set differences:

   $$f^{-1}(U\setminus V)=f^{-1}(U)\setminus f^{-1}(V)$$

3. Symmetric differences:

   $$f^{-1}(U△V)=f^{-1}(U)△f^{-1}(V)$$

In addition, for $X\subset A$ and $Y\subset B$, the inverse image satisfies the miscellaneous identities

1. ${({f|}_X)}^{-1}(Y)=X\cap f^{-1}(Y)$

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

3. $X\subset f^{-1}(f(X))$, with equality if $f$ is injective.

Title	inverse image
Canonical name	InverseImage
Date of creation	2013-03-22 11:51:58
Last modified on	2013-03-22 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 82-00
Classification	msc 83-00
Classification	msc 81-00
Synonym	preimage
Related topic	Mapping
Related topic	DirectImage