TeXipedia

complement

Denotes the complement of a set in mathematical notation, representing all elements not contained in a given set.

Overview

Essential in set theory and mathematical logic for expressing relationships between sets and their complements within a universal set.

  • Commonly used in probability theory to describe mutually exclusive events
  • Appears frequently in discrete mathematics and computer science for Boolean operations
  • Important in topology and analysis for describing open and closed sets
  • Often combined with other set operations like union and intersection in complex mathematical expressions

Examples

Set complement notation in set theory

A∁B={x∈U:x∉B}A \complement B = \{x \in U : x \notin B\} 
A \complement B = \{x \in U : x \notin B\}

Probability of complement event

P(A∁)=1−P(A)P(A \complement) = 1 - P(A) 
P(A \complement) = 1 - P(A)

Complement in relative to universal set

U∖A=A∁U \setminus A = A\complement 
U \setminus A = A\complement