the inverse image commutes with set operations

Theorem. Let $f$ be a mapping from $X$ to $Y$. If ${\{B_i\}}_{i\in I}$ is a (possibly uncountable) collection of subsets in $Y$, then the following relations hold for the inverse image:

1. $f^{-1}\left({\displaystyle \bigcup _{i\in I}}{B}_i\right)={\displaystyle \bigcup _{i\in I}}{f}^{-1}\left(B_i\right)$

2. $f^{-1}\left({\displaystyle \bigcap _{i\in I}}{B}_i\right)={\displaystyle \bigcap _{i\in I}}{f}^{-1}\left(B_i\right)$

If $A$ and $B$ are subsets in $Y$, then we also have:

1. For the set complement,

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

2. For the set difference,

   $$f^{-1}(A\setminus B)=f^{-1}(A)\setminus f^{-1}(B).$$

3. For the symmetric difference,

   $$f^{-1}(A△B)=f^{-1}(A)△f^{-1}(B).$$

Proof. For part (1), we have

$f^{-1}\left({\displaystyle \bigcup _{i\in I}}{B}_i\right)$ $=$ $\{x\in X\mid f(x)\in {\displaystyle \bigcup _{i\in I}}{B}_i\}$

$=$ $\{x\in X\mid f(x)\in B_i\text{for some}i\in I\}$

$=$ ${\displaystyle \bigcup _{i\in I}}\{x\in X\mid f(x)\in B_i\}$

$=$ ${\displaystyle \bigcup _{i\in I}}{f}^{-1}\left(B_i\right).$

Similarly, for part (2), we have

$f^{-1}\left({\displaystyle \bigcap _{i\in I}}{B}_i\right)$ $=$ $\{x\in X\mid f(x)\in {\displaystyle \bigcap _{i\in I}}{B}_i\}$

$=$ $\{x\in X\mid f(x)\in B_i\text{for all}i\in I\}$

$=$ ${\displaystyle \bigcap _{i\in I}}\{x\in X\mid f(x)\in B_i\}$

$=$ ${\displaystyle \bigcap _{i\in I}}{f}^{-1}\left(B_i\right).$

For the set complement, suppose $x\notin f^{-1}(A)$. This is equivalent to $f(x)\notin A$, or $f(x)\in A^{\mathrm{\complement }}$, which is equivalent to $x\in f^{-1}(A^{\mathrm{\complement }})$. Since the set difference $A\setminus B$ can be written as $A\cap B^c$, part (4) follows from parts (2) and (3). Similarly, since $A△B=(A\setminus B)\cup (B\setminus A)$, part (5) follows from parts (1) and (4). $\mathrm{\square }$

Title	the inverse image commutes with set operations
Canonical name	TheInverseImageCommutesWithSetOperations
Date of creation	2013-03-22 13:35:24
Last modified on	2013-03-22 13:35:24
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	11
Author	matte (1858)
Entry type	Proof
Classification	msc 03E20
Related topic	SetDifference