In set theory, the complement of a setA, often denoted by $A^{\complement }$ (or A′),^{[1]} is the set of elements not in A.^{[2]}
If A is the area colored red in this image…
… then the complement of A is everything else.
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 $B\setminus A,$ is the set of elements in B that are not in A.
Absolute complement
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Definition
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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]}$A^{\complement }=U\setminus A=\{x\in U:x\notin A\}.$
The absolute complement of A is usually denoted by $A^{\complement }$. Other notations include ${\overline {A}},A',$^{[2]}$\complement _{U}A,{\text{ and }}\complement A.$^{[4]}
Examples
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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.
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
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Definition
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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 $B\setminus A$ 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.
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.
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
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A binary relation$R$ is defined as a subset of a product of sets$X\times Y.$ The complementary relation${\bar {R}}$ is the set complement of $R$ in $X\times Y.$ The complement of relation $R$ can be written
${\bar {R}}\ =\ (X\times Y)\setminus R.$
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.
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 $\complement$ (as opposed to $C$) is produced by \complement. (It corresponds to the Unicode symbol U+2201∁COMPLEMENT.)
See also
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Algebra of sets – Identities and relationships involving sets
^The set in which the complement is considered is thus implicitly mentioned in an absolute complement, and explicitly mentioned in a relative complement.
^[1] Archived 2022-03-05 at the Wayback Machine The Comprehensive LaTeX Symbol List
References
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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.