Characteristic function

In set theory, the characteristic function or indicator function of a subset A of a set X is the function, often denoted χ_A or I_A, from X to the set {0,1} which takes the value 1 on elements of A and 0 otherwise.

We can express elementary set-theoretic operations in terms of characteristic functions:

• Empty set: $\chi _{\emptyset }=0;\,$
• Intersection: $\chi _{A\cap B}=\min\{\chi _{A},\chi _{B}\}=\chi _{A}\cdot \chi _{B};\,$
• Union: $\chi _{A\cup B}=\max\{\chi _{A},\chi _{B}\}=\chi _{A}+\chi _{B}-\chi _{A}\cdot \chi _{B};\,$
• Symmetric difference: $\chi _{A\bigtriangleup B}=\chi _{A}+\chi _{B}{\pmod {2}};\,$
• Inclusion: $A\subseteq B\Leftrightarrow \chi _{A}\leq \chi _{B}.\,$