inverse image

Let $f:A\longrightarrow B$ be a function, and let $U\subset B$ be a subset. The 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. 1.

Unions:

 $f^{-1}\left(\bigcup_{i\in I}U_{i}\right)=\bigcup_{i\in I}f^{-1}(U_{i})$
2. 2.
 $f^{-1}\left(\bigcap_{i\in I}U_{i}\right)=\bigcap_{i\in I}f^{-1}(U_{i})$

and for any subsets $U$ and $V$ of $B$, we have identities for

1. 3.
 $\left(f^{-1}(U)\right)^{\complement}=f^{-1}(U^{\complement})$
2. 4.
 $f^{-1}(U\setminus V)=f^{-1}(U)\setminus f^{-1}(V)$
3. 5.
 $f^{-1}(U\bigtriangleup V)=f^{-1}(U)\bigtriangleup f^{-1}(V)$

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

1. 6.

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

2. 7.

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

3. 8.

$X\subset f^{-1}(f(X))$, 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