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In set theory, the complement of a set A, often denoted by Ac (or A′), is the set of elements not in A. When all sets in the universe, i.e. all sets 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 [math]\displaystyle{ B \setminus A, }[/math] is the set of elements in B that are not in A.
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:^{[1]} [math]\displaystyle{ A^c = U \setminus A. }[/math]
Or formally: [math]\displaystyle{ A^c = \{ x \in U : x \notin A \}. }[/math]
The absolute complement of A is usually denoted by A^{c}. Other notations include [math]\displaystyle{ \overline A, A', }[/math]^{[2]} [math]\displaystyle{ \complement_U A, \text{ and } \complement A. }[/math]^{[3]}
Let A and B be two sets in a universe U. The following identities capture important properties of absolute complements:
De Morgan's laws:^{[4]}
Complement laws:^{[4]}
Involution or double complement law:
Relationships between relative and absolute complements:
Relationship with a set difference:
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,^{[4]} also termed the set difference of B and A,^{[5]} is the set of elements in B but not in A.
The relative complement of A in B is denoted [math]\displaystyle{ B \setminus A }[/math] according to the ISO 31-11 standard. It is sometimes written [math]\displaystyle{ B - A, }[/math] 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 [math]\displaystyle{ b - a, }[/math] where b is taken from B and a from A.
Formally: [math]\displaystyle{ B \setminus A = \{ x\in B : x \notin A \}. }[/math]
Let A, B, and C be three sets. The following identities capture notable properties of relative complements:
A binary relation [math]\displaystyle{ R }[/math] is defined as a subset of a product of sets [math]\displaystyle{ X \times Y. }[/math] The complementary relation [math]\displaystyle{ \bar{R} }[/math] is the set complement of [math]\displaystyle{ R }[/math] in [math]\displaystyle{ X \times Y. }[/math] The complement of relation [math]\displaystyle{ R }[/math] can be written [math]\displaystyle{ \bar{R} \ = \ (X \times Y) \setminus R. }[/math] Here, [math]\displaystyle{ R }[/math] is often viewed as a logical matrix with rows representing the elements of [math]\displaystyle{ X, }[/math] and columns elements of [math]\displaystyle{ Y. }[/math] The truth of [math]\displaystyle{ aRb }[/math] corresponds to 1 in row [math]\displaystyle{ a, }[/math] column [math]\displaystyle{ b. }[/math] Producing the complementary relation to [math]\displaystyle{ R }[/math] 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.
In the LaTeX typesetting language, the command \setminus
^{[6]} 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.
Some programming languages have sets among their builtin data structures. Such a data structure behaves as a finite set, that is, it consists of a finite number of data that are not specifically ordered, and may thus be considered as the elements of a set. In some cases, the elements are not necessary distinct, and the data structure codes multisets rather than sets. These programming languages have operators or functions for computing the complement and the set differences.
These operators may generally be applied also to data structures that are not really mathematical sets, such as ordered lists or arrays. It follows that some programming languages may have a function called set_difference
, even if they do not have any data structure for sets.