In set theory, the complement of a set refers to elements not in. When all sets under consideration are considered to be subsets of a given set, the absolute complement of is the set of elements in but not in. The relative complement of with respect to a set, also termed the set difference of and, written, is the set of elements in but not in.
Absolute complement
Definition
If is a set, then the absolute complement of is the set of elements not in, within a larger set that is implicitly defined. In other words, let be a set that contains all the elements under study; if it is no need to mention, either because it has been previously specified, or it is obvious and unique, then the absolute complement of is the relative complement of in : Formally: The absolute complement of is usually denoted by. Other notations include,,, and.
Assume that the universe is the standard 52-card deck. If the set is the suit of spades, then the complement of is the union of the suits of clubs, diamonds, and hearts. If the set is the union of the suits of clubs and diamonds, then the complement of is the union of the suits of hearts and spades.
Properties
Let and be two sets in a universe. The following identities capture important properties of absolute complements: The first two complement laws above show that if is a non-empty, proper subset of, then is a partition of.
Relative complement
Definition
If and are sets, then the relative complement of in, also termed the set difference of and, is the set of elements in but not in. The relative complement of in is denoted according to the ISO 31-11 standard. It is sometimes written, but this notation is ambiguous, as in some contexts it can be interpreted as the set of all elements, where is taken from and from. Formally:
Let,, and be three sets. The following identities capture notable properties of relative complements:
Complementary relation
A binary relationR is defined as a subset of a product of setsX × Y. The complementary relation is the set complement of R in X × Y. The complement of relation R can be written Often R is 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.
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. 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 have not any data structure for sets.
