Complement (set theory)

A circle filled with red inside a square. The area outside the circle is unfilled. The borders of both the circle and the square are black.

If

A

is the area colored red in this image…

An unfilled circle inside a square. The area inside the square not covered by the circle is filled with red. The borders of both the circle and the square are black.

… then the complement of

A

is everything else.

In set theory, the complement of a set $A$ , often denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^c} (or $A'$ ),^[1] is the set of elements not in $A$ .^[2]

When all elements in the universe, i.e. all elements under consideration, are considered to be members 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B \setminus A,} is the set of elements in $B$ that are not in $A$ .

Absolute complement

File:Venn10.svg

The absolute complement of the white disc is the red region

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]^{[lower-alpha 1]} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^c= U \setminus A = \{ x \in U : x \notin A \}.}

The absolute complement of $A$ is usually denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^c} .^[3] Other notations include Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline A } ,^[4] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A',} ^[2] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \complement_U A, \text{ and } \complement A.} ^[5]

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:^[3]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(A \cup B \right)^c= A^c \cap B^c.}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(A \cap B \right)^c = A^c \cup B^c.}

Complement laws:^[3]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cup A^c = U.}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cap A^c = \empty .}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \empty^c = U.}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U^c = \empty.}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(A^c\right)^c = A.}

Relationships between relative and absolute complements:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \setminus B = A \cap B^c.}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A \setminus B)^c = A^c \cup B = A^c \cup (B \cap A).}

Relationship with a set difference:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 ${A, A ∁} Template:Null$ is a partition of $U$ .

Relative complement

Definition

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 $A$ .

File:Relative compliment.svg

The relative complement of

A

in

B

: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B \cap A^c = B \setminus A}

The relative complement of $A$ in $B$ is denoted Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B \setminus A} according to the ISO 31-11 standard. It is sometimes written Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b - a,} where $b$ is taken from $B$ and $a$ from $A$ .

Formally: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B \setminus A = \{ x\in B : x \notin A \}.}

Examples

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ 1, 2, 3\} \setminus \{ 2,3,4\} = \{ 1 \}.}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ 2, 3, 4 \} \setminus \{ 1,2,3 \} = \{ 4 \} .}
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} is the set of real numbers and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Q}} is the set of rational numbers, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}\setminus\mathbb{Q}} is the set of irrational numbers.

Properties

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C \setminus (A \cap B) = (C \setminus A) \cup (C \setminus B).}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C \setminus (A \cup B) = (C \setminus A) \cap (C \setminus B).}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C \setminus (B \setminus A) = (C \cap A) \cup (C \setminus B),}
with the important special case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C \setminus (C \setminus A) = (C \cap A)} demonstrating that intersection can be expressed using only the relative complement operation.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (B \setminus A) \cap C = (B \cap C) \setminus A = B \cap (C \setminus A).}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (B \setminus A) \cup C = (B \cup C) \setminus (A \setminus C).}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \setminus A = \emptyset.}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \empty \setminus A = \empty.}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \setminus \empty = A.}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \setminus U = \empty.}
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\subset B} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C\setminus A\supset C\setminus B} .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \supseteq B \setminus C} is equivalent to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C \supseteq B \setminus A} .

Complementary relation

A binary relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is defined as a subset of a product of sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \times Y.} The complementary relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{R}} is the set complement of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \times Y.} The complement of relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} can be written Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{R} \ = \ (X \times Y) \setminus R.} Here, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is often viewed as a logical matrix with rows representing the elements of $X,$ and columns elements of $Y.$ The truth of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle aRb} corresponds to 1 in row Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a,} column Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b.} Producing the complementary relation to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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^[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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \complement} (as opposed to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} ) is produced by \complement. (It corresponds to the Unicode symbol U+2201 ∁ COMPLEMENT.)

Footnotes

↑ The set in which the complement is considered is thus implicitly mentioned in an absolute complement, and explicitly mentioned in a relative complement.^[3]

Notes

↑ "Complement and Set Difference". web.mnstate.edu. Retrieved 2020-09-04.
↑ ^2.0 ^2.1 "Complement (set) Definition (Illustrated Mathematics Dictionary)". www.mathsisfun.com. Retrieved 2020-09-04.
↑ ^3.0 ^3.1 ^3.2 ^3.3 ^3.4 ^3.5 Halmos 1960, p. 17.
↑ Stoll 1979, p. 19.
↑ Bourbaki 1970, p. E II.6.
↑ Devlin 1979, p. 6.
↑ [1] Archived 2022-03-05 at the Wayback Machine The Comprehensive LaTeX Symbol List

References

Bourbaki, N. (1970). Théorie des ensembles (in French). Paris: Hermann. ISBN 978-3-540-34034-8.
Devlin, Keith J. (1979). Fundamentals of contemporary set theory. Universitext. Springer. ISBN 0-387-90441-7. Zbl 0407.04003.
Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van Nostrand Company. ISBN 9780442030643. Zbl 0087.04403.
Stoll, Robert R. (1979). Set Theory and Logic. Mineola, N.Y.: Dover Publications. ISBN 0-486-63829-4.

External links

Template:Set theory Template:Mathematical logic

[4] The set in which the complement is considered is thus implicitly mentioned in an absolute complement, and explicitly mentioned in a relative complement.^[3]

[1] "Complement and Set Difference". web.mnstate.edu. Retrieved 2020-09-04.

[:1-2] 2.0 ^2.1 "Complement (set) Definition (Illustrated Mathematics Dictionary)". www.mathsisfun.com. Retrieved 2020-09-04.

[FOOTNOTEHalmos1960[httpsbooksgooglecombooksidlgdJDgAAQBAJpgPA17_17]-3] 3.0 ^3.1 ^3.2 ^3.3 ^3.4 ^3.5 Halmos 1960, p. 17.

[FOOTNOTEStoll1979[httpsarchiveorgdetailssettheorylogiced00robepage19_19]-5] Stoll 1979, p. 19.

[Bou-6] Bourbaki 1970, p. E II.6.

[FOOTNOTEDevlin19796-7] Devlin 1979, p. 6.

[The_Comprehensive_LaTeX_Symbol_List-8] [1] Archived 2022-03-05 at the Wayback Machine The Comprehensive LaTeX Symbol List

[1]

[2]

[3]

[lower-alpha 1]

[4]

[5]

[6]

[7]

Complement (set theory)

Contents

Absolute complement

Definition

Examples

Properties

Relative complement

Definition

Examples

Properties

Complementary relation

LaTeX notation

See also

Footnotes

Notes

References

External links

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Complement (set theory)

Absolute complement

Definition

Examples

Properties

Relative complement

Definition

Examples

Properties

Complementary relation

LaTeX notation

See also

Footnotes

Notes

References

External links

Navigation menu

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