Definition

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:^[3]

$${\displaystyle A^{\complement }=U\setminus A=\{x\in U:x\notin A\}.}$$

The absolute complement of A is usually denoted by ${\displaystyle A^{\complement }}$. Other notations include ${\displaystyle {\overline {A}},A',}$^[2] ${\displaystyle \complement _{U}A,{\text{ and }}\complement A.}$^[4]

Examples

• 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.
• When the universe is the universe of sets described in formalized set theory, the absolute complement of a set is generally not itself a set, but rather a proper class. For more info, see universal set.

Properties

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

De Morgan's laws:^[5]

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

Complement laws:^[5]

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

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

Involution or double complement law:

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

Relationships between relative and absolute complements:

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

Relationship with a set difference:

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

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

Definition

If A and B are sets, then the relative complement of A in B,^[5] also termed the set difference of B and A,^[6] is the set of elements in B but not in A.

The relative complement of A in B is denoted ${\displaystyle B\setminus A}$ according to the ISO 31-11 standard. It is sometimes written ${\displaystyle B-A,}$ but this notation is ambiguous, as in some contexts (for example, Minkowski set operations in functional analysis) it can be interpreted as the set of all elements ${\displaystyle b-a,}$ where b is taken from B and a from A.

Formally:

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

Examples

• ${\displaystyle \{1,2,3\}\setminus \{2,3,4\}=\{1\}.}$
• ${\displaystyle \{2,3,4\}\setminus \{1,2,3\}=\{4\}.}$
• 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.

Properties

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.

• ${\displaystyle (B\setminus A)\cap C=(B\cap C)\setminus A=B\cap (C\setminus A).}$
• ${\displaystyle (B\setminus A)\cup C=(B\cup C)\setminus (A\setminus C).}$
• ${\displaystyle A\setminus A=\emptyset .}$
• ${\displaystyle \emptyset \setminus A=\emptyset .}$
• ${\displaystyle A\setminus \emptyset =A.}$
• ${\displaystyle A\setminus U=\emptyset .}$
• If ${\displaystyle A\subset B}$, then ${\displaystyle C\setminus A\supset C\setminus B}$.
• ${\displaystyle A\supseteq B\setminus C}$ is equivalent to ${\displaystyle C\supseteq B\setminus A}$.