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Equinumerosity has the characteristic properties of an equivalence relation (reflexivity, symmetry, and transitivity): [1] Reflexivity Given a set A, the identity function on A is a bijection from A to itself, showing that every set A is equinumerous to itself: A ~ A. Symmetry
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 X {\displaystyle X} 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.
Reflexivity: for every a, one has a = a. Symmetry: for every a and b, if a = b, then b = a. Transitivity: for every a, b, and c, if a = b and b = c, then a = c. [9] [10] Substitution: Informally, this just means that if a = b, then a can replace b in any mathematical expression or formula without changing its meaning.
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.
Generalized to stochastic versions (stochastic transitivity), the study of transitivity finds applications of in decision theory, psychometrics and utility models. [19] A quasitransitive relation is another generalization; [5] it is required to be transitive only on its non-symmetric part. Such relations are used in social choice theory or ...
A ternary equivalence relation is symmetric, reflexive, and transitive, where those terms are meant in the sense defined below. The classic example is the relation of collinearity among three points in Euclidean space .
Formally, a relation on a set is a PER if it holds for all ,, that: . if , then (symmetry); if and , then (transitivity); Another more intuitive definition is that on a set is a PER if there is some subset of such that and is an equivalence relation on .