Complement (set theory)

In set theory, the complement of a set Template:Mvar, often denoted by $A^c$ (or A′), is the set of elements not in Template:Mvar.

When all elements in the universe, i.e. all elements under consideration, are considered to be members of a given set Template:Mvar, the absolute complement of Template:Mvar is the set of elements in Template:Mvar that are not in Template:Mvar.

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

Absolute complement

File:Venn10.svg

The absolute complement of the white disc is the red region

Definition

If Template:Mvar is a set, then the absolute complement of Template:Mvar (or simply the complement of Template:Mvar) is the set of elements not in Template:Mvar (within a larger set that is implicitly defined). In other words, let Template:Mvar be a set that contains all the elements under study; if there is no need to mention Template:Mvar, either because it has been previously specified, or it is obvious and unique, then the absolute complement of Template:Mvar is the relative complement of Template:Mvar in Template:Mvar: $$A^c=U\setminus A=\{x\in U:x\notin A\}.$$

The absolute complement of Template:Mvar is usually denoted by $A^c$. Other notations include $\bar{A},A^{\prime },$ ${\complement }_{U}A,\text{ and }\complement A.$

Examples

• Assume that the universe is the set of integers. If Template:Mvar is the set of odd numbers, then the complement of Template:Mvar is the set of even numbers. If Template:Mvar is the set of multiples of 3, then the complement of Template:Mvar 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 Template:Mvar is the suit of spades, then the complement of Template:Mvar is the union of the suits of clubs, diamonds, and hearts. If the set Template:Mvar is the union of the suits of clubs and diamonds, then the complement of Template:Mvar 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 Template:Mvar and Template:Mvar be two sets in a universe Template:Mvar. The following identities capture important properties of absolute complements:

De Morgan's laws:

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

Complement laws:

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

  (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^∁} is a partition of U.

Relative complement

Definition

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

File:Relative compliment.svg

The relative complement of A in B: $B\cap A^c=B\setminus A$

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

Examples

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

Properties

Let A, B, and C be three sets in a universe Template:Mvar. 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.

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

Complementary relation

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

Together with composition of relations and converse relations, complementary relations and the algebra of sets are the elementary operations of the calculus of relations.

LaTeX notation

In the LaTeX typesetting language, the command \setminus 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 $\complement$ (as opposed to $C$) is produced by \complement. (It corresponds to the Unicode symbol Template:Unichar.)

See also

• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link