Complement (set theory)

In set theory, the complement of a set A, often denoted by ${\displaystyle A^{\complement }}$ (or A′),^[1] is the set of elements not in A.^[2]

When all elements in the

of a given set U, the absolute complement of A is the set of elements in U that are not in A.

The relative complement of A with respect to a set B, also termed the set difference of B and A, written ${\displaystyle B\setminus A,}$ is the set of elements in B that are not in A.

Absolute complement

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]

• 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
  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,

of U.

Relative complement

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

• ${\displaystyle \{1,2,3\}\setminus \{2,3,4\}=\{1\}.}$
• ${\displaystyle \{2,3,4\}\setminus \{1,2,3\}=\{4\}.}$
• 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 in a universe U. 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 ${\displaystyle C\supseteq B\setminus A}$.

A binary relation ${\displaystyle R}$ is defined as a subset of a

${\displaystyle X\times Y.}$ The complementary relation ${\displaystyle {\bar {R}}}$ is the set complement of ${\displaystyle R}$ in ${\displaystyle X\times Y.}$ The complement of relation ${\displaystyle R}$ can be written $${\displaystyle {\bar {R}}\ =\ (X\times Y)\setminus R.}$$ Here, ${\displaystyle R}$ is often viewed as a logical matrix with rows representing the elements of ${\displaystyle X,}$ and columns elements of ${\displaystyle Y.}$ The truth of ${\displaystyle aRb}$ corresponds to 1 in row ${\displaystyle a,}$ column ${\displaystyle b.}$ Producing the complementary relation to ${\displaystyle R}$ then corresponds to switching all 1s to 0s, and 0s to 1s for the logical matrix of the complement.

Together with

.

In the LaTeX typesetting language, the command \setminus^[7] is usually used for rendering a set difference symbol, which is similar to a backslash symbol. When rendered, the \setminus command looks identical to \backslash, except that it has a little more space in front and behind the slash, akin to the LaTeX sequence \mathbin{\backslash}. A variant \smallsetminus is available in the amssymb package, but this symbol is not included separately in Unicode. The symbol ${\displaystyle \complement }$ (as opposed to ${\displaystyle C}$) is produced by \complement. (It corresponds to the Unicode symbol U+2201 ∁ COMPLEMENT.)

• Algebra of sets – Identities and relationships involving sets
• Intersection (set theory) – Set of elements common to all of some sets
• List of set identities and relations – Equalities for combinations of sets
• Naive set theory – Informal set theories
• Symmetric difference – Elements in exactly one of two sets
• Union (set theory) – Set of elements in any of some sets

• ISBN 978-3-540-34034-8
  .
• Zbl 0407.04003
  .
• Zbl 0087.04403
  .

External links

• Weisstein, Eric W. "Complement". MathWorld.
• Weisstein, Eric W. "Complement Set". MathWorld.