GeezerVison

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][a] $A^c=U\setminus A=\{x\in U:x\notin A\}.$

The absolute complement of A is usually denoted by $A^c$.^[3] Other notations include $\bar{A}$,^[4] $A^{\prime },$^[2] ${\complement }_{U}A,\text{ and }\complement A.$^[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.
• 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.

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

De Morgan's laws:^[3]

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

Complement laws:^[3]

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

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

Involution or double complement law:

• ${\left(A^c\right)}^c=A.$

Relationships between relative and absolute complements:

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

Relationship with a set difference:

• $A^c\setminus B^c=B\setminus A.$

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,^[3] 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 $B\setminus A$ according to the ISO 31-11 standard. It is sometimes written $B-A,$ but this notation can be ambiguous, as in some contexts (for example, Minkowski set operations in functional analysis) it can be interpreted as the set of all elements $b-a,$ where b is taken from B and a from A.

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

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

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

• $C\setminus (A\cap B)=(C\setminus A)\cup (C\setminus B).$
• $C\setminus (A\cup B)=(C\setminus A)\cap (C\setminus B).$
• $C\setminus (B\setminus A)=(C\cap A)\cup (C\setminus B),$
  with the important special case $C\setminus (C\setminus A)=(C\cap A)$ demonstrating that intersection can be expressed using only the relative complement operation.
• $(B\setminus A)\cap C=(B\cap C)\setminus A=B\cap (C\setminus A).$
• $(B\setminus A)\cup C=(B\cup C)\setminus (A\setminus C).$
• $A\setminus A=\mathrm{\emptyset }.$
• $\mathrm{\emptyset }\setminus A=\mathrm{\emptyset }.$
• $A\setminus \mathrm{\emptyset }=A.$
• $A\setminus U=\mathrm{\emptyset }.$
• If $A\subset B$, then $C\setminus A\supset C\setminus B$.
• $A\supseteq B\setminus C$ is equivalent to $C\supseteq B\setminus A$.