Complement (set theory)

Lore & Background

The absolute complement of a set A, denoted A^c, is defined as U ∖ A = {x ∈ U : x ∉ A}, where U is the universal set containing all elements under study. Other notations include A̅, A′, ∁_U A, and ∁ A. For example, if the universe is the integers and A is the set of odd numbers, then the complement of A is the set of even numbers. If the universe is a standard 52-card deck and A is the suit of spades, then the complement of A is the union of clubs, diamonds, and hearts.

Reader's Guide

The complement operation is fundamental in set theory, providing a way to express negation. Key properties include De Morgan's laws: (A ∪ B)^c = A^c ∩ B^c and (A ∩ B)^c = A^c ∪ B^c. Complement laws state A ∪ A^c = U, A ∩ A^c = ∅, ∅^c = U, U^c = ∅, and if A ⊆ B then B^c ⊆ A^c. The double complement law gives (A^c)^c = A. The relative complement B ∖ A = {x ∈ B : x ∉ A} relates to absolute complement via A ∖ B = A ∩ B^c. These identities are essential for manipulating set expressions and form the basis for Boolean algebra and logic.

Did You Know?

• The absolute complement of a set A is defined as U ∖ A, where U is the universal set.
• De Morgan's laws for complements are (A ∪ B)^c = A^c ∩ B^c and (A ∩ B)^c = A^c ∪ B^c.
• The relative complement B ∖ A is also called the set difference of B and A.
• If the universe is the universe of sets in formalized set theory, the absolute complement of a set is generally a proper class, not a set.