Definition 5.5.1
Let be a set and be a subset of . The characteristic function of (also called the indicator function of the subset of is the mapping
defined by

Note that two subsets
and of are equal if, and only if,
Denote by
the mapping which assigns the number to every element of .

Proposition 5.5.2
Let and be two subsets of a set ; we denote by and their respective characteristic functions. Then the following relations hold:

In symmetric difference, we proved already the following proposition, using either truth tables or set properties (see prop associativity of symmetric difference 1). We prove the proposition here once more, using characteristic functions.

Proposition 5.5.3
Let , and be three sets. Then the following relation holds:

Proof.
For, modulo 2,

Pure algebraists will note that this property together with other properties proven in Chapter 3 endow the power set of a given set with a structure of an abelian group. Let be nay set. Then we have:

Closure: Given
,
;

Associativity: for any in
,
;

Commutativity: for any in
,
;

Identity: for any in
,
for all
;

Inverse:
for all
.

Proposition 5.5.4 (De Morgan's Laws revisited)
Let and be two subsets of the set . Then the following relations hold: