characteristic function

Definition Suppose $A$ is a subset of a set $X$ . Then thefunction

\\chi_{A}(x)=\\begin{cases}1,&\\text{when }x\\in A,\\\\0,&\\text{when }x\\in X\\setminus A\\end{cases}

is the characteristic function for $A$ .

0.0.1 Properties

Suppose $A, B$ are subsets of a set $X$ .

1.
For set intersections and set unions, we have
$\\displaystyle\\chi_{A\\cap B}$ $\\displaystyle=$ $\\displaystyle\\chi_{A}\\chi_{B},$
$\\displaystyle\\chi_{A\\cup B}$ $\\displaystyle=$ $\\displaystyle\\chi_{A}+\\chi_{B}-\\chi_{A\\cap B},$
$\\displaystyle\\chi_{A\\cap B}$ $\\displaystyle=$ $\\displaystyle\\min(\\chi_{A},\\chi_{B}),$
$\\displaystyle\\chi_{A\\cup B}$ $\\displaystyle=$ $\\displaystyle\\max(\\chi_{A},\\chi_{B}).$
2.
For the symmetric difference,
$\\chi_{A\\bigtriangleup B}=\\chi_{A}+\\chi_{B}-2\\chi_{A\\cap B}.$
3.
For the set complement,
$\\chi_{A^{\\complement}}=1-\\chi_{A}.$

A synonym for characteristic function is indicator function[1].

1 G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.