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.
IntransitivityIn mathematics, intransitivity (sometimes called nontransitivity) is a property of binary relations that are not transitive relations. This may include any relation that is not transitive, or the stronger property of antitransitivity, which describes a relation that is never transitive. A relation is transitive if, whenever it relates some A to some B, and that B to some C, it also relates that A to that C.
Homogeneous relationIn mathematics, a homogeneous relation (also called endorelation) on a set X is a binary relation between X and itself, i.e. it is a subset of the Cartesian product X × X. This is commonly phrased as "a relation on X" or "a (binary) relation over X". An example of a homogeneous relation is the relation of kinship, where the relation is between people. Common types of endorelations include orders, graphs, and equivalences. Specialized studies of order theory and graph theory have developed understanding of endorelations.
Transitive relationIn mathematics, a relation R on a set X is transitive if, for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c. Each partial order as well as each equivalence relation needs to be transitive. A homogeneous relation R on the set X is a transitive relation if, for all a, b, c ∈ X, if a R b and b R c, then a R c. Or in terms of first-order logic: where a R b is the infix notation for (a, b) ∈ R. As a non-mathematical example, the relation "is an ancestor of" is transitive.
Inequality (mathematics)In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different notations used to represent different kinds of inequalities: The notation a < b means that a is less than b. The notation a > b means that a is greater than b. In either case, a is not equal to b. These relations are known as strict inequalities, meaning that a is strictly less than or strictly greater than b.
Law of trichotomyIn mathematics, the law of trichotomy states that every real number is either positive, negative, or zero. More generally, a binary relation R on a set X is trichotomous if for all x and y in X, exactly one of xRy, yRx and x = y holds. Writing R as
Connected relationIn mathematics, a relation on a set is called connected or complete or total if it relates (or "compares") all pairs of elements of the set in one direction or the other while it is called strongly connected if it relates pairs of elements. As described in the terminology section below, the terminology for these properties is not uniform. This notion of "total" should not be confused with that of a total relation in the sense that for all there is a so that (see serial relation).
Symmetric relationA symmetric relation is a type of binary relation. An example is the relation "is equal to", because if a = b is true then b = a is also true. Formally, a binary relation R over a set X is symmetric if: where the notation means that . If RT represents the converse of R, then R is symmetric if and only if R = RT. Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation. "is equal to" (equality) (whereas "is less than" is not symmetric) "is comparable to", for elements of a partially ordered set ".
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.
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.
