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In mathematics, a ternary equivalence relation is a kind of ternary relation analogous to a binary equivalence relation.A ternary equivalence relation is symmetric, reflexive, and transitive, where those terms are meant in the sense defined below.
A reflexive and symmetric relation is a dependency relation (if finite), and a tolerance relation if infinite. A preorder is reflexive and transitive. A congruence relation is an equivalence relation whose domain is also the underlying set for an algebraic structure, and which respects the additional structure.
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.
This last property is often called co-transitivity or comparison. The complement of an apartness relation is an equivalence relation, as the above three conditions become reflexivity, symmetry, and transitivity. If this equivalence relation is in fact equality, then the apartness relation is called tight.
It is defined on a set as a binary relation that satisfies the three properties: reflexivity, symmetry, and transitivity. Reflexivity means that every element in X {\displaystyle X} is equivalent to itself ( a ∼ a {\displaystyle a\sim a} for all a ∈ X {\displaystyle a\in X} ).
However, there is a formula for finding the number of relations that are simultaneously reflexive, symmetric, and transitive – in other words, equivalence relations – (sequence A000110 in the OEIS), those that are symmetric and transitive, those that are symmetric, transitive, and antisymmetric, and those that are total, transitive, and ...
Symmetric, transitive, and reflexive relations are distinguished by their structural features. Metaphysical difficulties like the question of where relations are located lie at the center of discussions of their ontological status. Eliminativism is the thesis that relations are mental abstractions that are not a part of external reality.
A relation that is reflexive, symmetric, and transitive. It is also a relation that is symmetric, transitive, and serial, since these properties imply reflexivity. Orderings: Partial order A relation that is reflexive, antisymmetric, and transitive. Strict partial order A relation that is irreflexive, asymmetric, and transitive. Total order