Antisymmetric relationIn mathematics, a binary relation on a set is antisymmetric if there is no pair of distinct elements of each of which is related by to the other. More formally, is antisymmetric precisely if for all or equivalently, The definition of antisymmetry says nothing about whether actually holds or not for any . An antisymmetric relation on a set may be reflexive (that is, for all ), irreflexive (that is, for no ), or neither reflexive nor irreflexive. A relation is asymmetric if and only if it is both antisymmetric and irreflexive.
Asymmetric relationIn mathematics, an asymmetric relation is a binary relation on a set where for all if is related to then is not related to A binary relation on is any subset of Given write if and only if which means that is shorthand for The expression is read as " is related to by " The binary relation is called if for all if is true then is false; that is, if then This can be written in the notation of first-order logic as A logically equivalent definition is: for all at least one of and is , which in first-order logic c
Reflexive relationIn mathematics, a binary relation R on a set X is reflexive if it relates every element of X to itself. An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations.
Power setIn mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The powerset of S is variously denoted as P(S), P(S), P(S), , , or 2S. The notation 2S, meaning the set of all functions from S to a given set of two elements (e.g.
Lattice (order)A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the power set of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection.
Empty setIn mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. 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. Many possible properties of sets are vacuously true for the empty set. Any set other than the empty set is called non-empty. In some textbooks and popularizations, the empty set is referred to as the "null set".
Finite setIn mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the cardinality (or the cardinal number) of the set. A set that is not a finite set is called an infinite set.
Universal setIn set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory include a universal set. Many set theories do not allow for the existence of a universal set. There are several different arguments for its non-existence, based on different choices of axioms for set theory. In Zermelo–Fraenkel set theory, the axiom of regularity and axiom of pairing prevent any set from containing itself.
Converse relationIn mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent of'. In formal terms, if and are sets and is a relation from to then is the relation defined so that if and only if In set-builder notation, The notation is analogous with that for an inverse function. Although many functions do not have an inverse, every relation does have a unique converse.
Complete partial orderIn mathematics, the phrase complete partial order is variously used to refer to at least three similar, but distinct, classes of partially ordered sets, characterized by particular completeness properties. Complete partial orders play a central role in theoretical computer science: in denotational semantics and domain theory. A complete partial order, abbreviated cpo, can refer to any of the following concepts depending on context. A partially ordered set is a directed-complete partial order (dcpo) if each of its directed subsets has a supremum.
Fuzzy setIn mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined a more general kind of structure called an L-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are now used throughout fuzzy mathematics and have applications in areas such as linguistics , decision-making , and clustering , are special cases of L-relations when L is the unit interval [0, 1].
Naive set theoryNaive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and suffices for the everyday use of set theory concepts in contemporary mathematics.
