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Set (abstract data type) In computer science, a set is an abstract data type that can store unique values, without any particular order. It is a computer implementation of the mathematical concept of a finite set. Unlike most other collection types, rather than retrieving a specific element from a set, one typically tests a value for membership ...
The set of all homogeneous relations () over a set is the power set which is a Boolean algebra augmented with the involution of mapping of a relation to its converse relation. Considering composition of relations as a binary operation on B ( X ) {\displaystyle {\mathcal {B}}(X)} , it forms a semigroup with involution .
The empty set is the set containing no elements. In mathematics, the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. [1] Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced.
the empty set is an extended binary tree; if T 1 and T 2 are extended binary trees, then denote by T 1 • T 2 the extended binary tree obtained by adding a root r connected to the left to T 1 and to the right to T 2 [clarification needed where did the 'r' go in the 'T 1 • T 2 ' symbol] by adding edges when these sub-trees are non-empty.
In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is ...
The term complete partial order, abbreviated cpo, has several possible meanings depending on context. A partially ordered set is a directed-complete partial order (dcpo) if each of its directed subsets has a supremum. (A subset of a partial order is directed if it is non-empty and every pair of elements has an upper bound in the subset.)
In computer science, a tree is a widely used abstract data type that represents a hierarchical tree structure with a set of connected nodes. Each node in the tree can be connected to many children (depending on the type of tree), but must be connected to exactly one parent, [1] except for the root node, which has no parent (i.e., the root node ...
A fuzzy set is a pair (,) where is a set (often required to be non-empty) and : [,] a membership function. The reference set (sometimes denoted by or ) is called universe of discourse, and for each , the value () is called the grade of membership of in (,).