In mathematics, a relation on a set is called connected or complete or total if it relates (or "compares") all distinct pairs of elements of the set in...

a relation denotes some kind of relationship between two objects in a set, which may or may not hold. As an example, "is less than" is a relation on...

which are connected by roads. Let R be the relation on towns where (A, B) ∈ R if there is a road directly linking town A and town B. This relation need not...

Connected and disconnected subspaces of R² In topology and related branches of mathematics, a connected space is a topological space that cannot be represented...

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...

binary relation associates some elements of one set called the domain with some elements of another set called the codomain. Precisely, a binary relation over...

of a locally connected space. As an example, the notion of connectedness im kleinen at a point and its relation to local connectedness will be considered...

relation was also used by B. A. Bernstein for an article showing that particular common axioms in order theory are nearly incompatible: connectedness...

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...

which any two elements are comparable. That is, a total order is a binary relation ≤ {\displaystyle \leq } on some set X {\displaystyle X} , which satisfies...

complete partial order (q.v.) with least element. Complete relation. Synonym for Connected relation. Complete semilattice. The notion of a complete semilattice...

partial order without incomparable pairs Total relation, which may also mean connected relation (a binary relation in which any two elements are comparable)...

In mathematics, a binary relation R {\displaystyle R} on a set X {\displaystyle X} is antisymmetric if there is no pair of distinct elements of X {\displaystyle...

mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in...

In mathematics, a binary relation R is called well-founded (or wellfounded or foundational) on a set or, more generally, a class X if every non-empty subset...

was first used in this sense by J. J. Sylvester in 1878 due to a direct relation between mathematics and chemical structure (what he called a chemico-graphical...

relation of being strongly connected is an equivalence relation, and the induced subgraphs of its equivalence classes are called strongly connected components...

relation is always quasitransitive, as is a left Euclidean relation. A connected right Euclidean relation is always transitive; and so is a connected...

In mathematics, an asymmetric relation is a binary relation R {\displaystyle R} on a set X {\displaystyle X} where for all a , b ∈ X , {\displaystyle...

A symmetric relation is a type of binary relation. Formally, a binary relation R over a set X is symmetric if: ∀ a , b ∈ X ( a R b ⇔ b R a ) , {\displaystyle...

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'...

orderings on a given finite set is not in general connected by moves that add or remove a single order relation to or from a given ordering. For instance, for...

types of binary relation. One specific variation of weak ordering, a total preorder (= a connected, reflexive and transitive relation), is also sometimes...

the way train cars are connected on a track. When a reader or listener "loses the train of thought" (i.e., loses the relation between consecutive sentences...

the transitive closure R+ of a homogeneous binary relation R on a set X is the smallest relation on X that contains R and is transitive. For finite sets...

mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. The name preorder is meant to suggest...

between a relation and its relata is made by the fact that instantiates the relation without the need for a second relation. A closely connected explanation...

the subset-inclusion relation on some collection of objects Mereology – Study of parts and the wholes they form Region – Connected open subset of a topological...

pair is comparable. Formally, a partial order is a homogeneous binary relation that is reflexive, antisymmetric, and transitive. A partially ordered set...

be true: x<y, x=y, y<x. A relation is trichotomous if, and only if, it is asymmetric and connected. If a trichotomous relation is also transitive, then...