Complement (set theory)

If A is a set, then the absolute complement of A (or simply the complement of A) is the set of elements not in A (within a larger set that is implicitly defined). In other words, let U be a set that contains all the elements under study; if there is no need to mention U, either because it has been previously specified, or it is obvious and unique, then the absolute complement of A is the relative complement of A in U:[4]

${\displaystyle A^{c}=U-A}$.

Or formally:

${\displaystyle A^{c}=\{x\in U\mid x\notin A\}.}$

The absolute complement of A is usually denoted by A^c.[1] Other notations include ${\displaystyle {\overline {A}}}$, ${\displaystyle A'}$,[3] ${\displaystyle \complement _{U}A}$, and ${\displaystyle \complement A}$.[5]

• Assume that the universe is the set of integers. If A is the set of odd numbers, then the complement of A is the set of even numbers. If B is the set of multiples of 3, then the complement of B is the set of numbers congruent to 1 or 2 modulo 3 (or, in simpler terms, the integers that are not multiples of 3).
• Assume that the universe is the standard 52-card deck. If the set A is the suit of spades, then the complement of A is the union of the suits of clubs, diamonds, and hearts. If the set B is the union of the suits of clubs and diamonds, then the complement of B is the union of the suits of hearts and spades.

Let A and B be two sets in a universe U. The following identities capture important properties of absolute complements:

De Morgan's laws:[6]

• ${\displaystyle \left(A\cup B\right)^{c}=A^{c}\cap B^{c}.}$
• ${\displaystyle \left(A\cap B\right)^{c}=A^{c}\cup B^{c}.}$

Complement laws:[6]

• ${\displaystyle A\cup A^{c}=U.}$
• ${\displaystyle A\cap A^{c}=\varnothing .}$
• ${\displaystyle \varnothing ^{c}=U.}$
• ${\displaystyle U^{c}=\varnothing .}$
• ${\displaystyle {\text{If }}A\subseteq B{\text{, then }}B^{c}\subseteq A^{c}.}$

  (this follows from the equivalence of a conditional with its contrapositive).

Involution or double complement law:

• ${\displaystyle \left(A^{c}\right)^{c}=A.}$

Relationships between relative and absolute complements:

• ${\displaystyle A\setminus B=A\cap B^{c}.}$
• ${\displaystyle (A\setminus B)^{c}=A^{c}\cup B.}$

Relationship with set difference:

• ${\displaystyle A^{c}\setminus B^{c}=B\setminus A.}$

The first two complement laws above show that if A is a non-empty, proper subset of U, then {A, A^c} is a partition of U.

If A and B are sets, then the relative complement of A in B,[6] also termed the set difference of B and A,[7] is the set of elements in B but not in A.

The relative complement of A (left circle) in B (right circle): ${\displaystyle B\cap A^{c}=B\setminus A}$

The relative complement of A in B is denoted B ∖ A according to the ISO 31-11 standard. It is sometimes written B − A,[1] but this notation is ambiguous, as in some contexts it can be interpreted as the set of all elements b − a, where b is taken from B and a from A.

Formally:

${\displaystyle B\setminus A=\{x\in B\mid x\notin A\}.}$

• ${\displaystyle \{1,2,3\}\setminus \{2,3,4\}=\{1\}}$.
• ${\displaystyle \{2,3,4\}\setminus \{1,2,3\}=\{4\}}$.
• If ${\displaystyle \mathbb {R} }$ is the set of real numbers and ${\displaystyle \mathbb {Q} }$ is the set of rational numbers, then ${\displaystyle \mathbb {R} \setminus \mathbb {Q} }$ is the set of irrational numbers.

Let A, B, and C be three sets. The following identities capture notable properties of relative complements:

• ${\displaystyle C\setminus (A\cap B)=(C\setminus A)\cup (C\setminus B)}$.
• ${\displaystyle C\setminus (A\cup B)=(C\setminus A)\cap (C\setminus B)}$.
• ${\displaystyle C\setminus (B\setminus A)=(C\cap A)\cup (C\setminus B)}$,

  with the important special case ${\displaystyle C\setminus (C\setminus A)=(C\cap A)}$ demonstrating that intersection can be expressed using only the relative complement operation.

• ${\displaystyle (B\setminus A)\cap C=(B\cap C)\setminus A=B\cap (C\setminus A)}$.
• ${\displaystyle (B\setminus A)\cup C=(B\cup C)\setminus (A\setminus C)}$.
• ${\displaystyle A\setminus A=\emptyset }$.
• ${\displaystyle \emptyset \setminus A=\emptyset }$.
• ${\displaystyle A\setminus \emptyset =A}$.
• ${\displaystyle A\setminus U=\emptyset }$.