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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.
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.
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.
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 .
Reflexivity For every a, one has a = a. [11] [12] Symmetry For every a and b, if a = b, then b = a. [11] [12] Transitivity For every a, b, and c, if a = b and b = c, then a = c. [11] [12] Substitution Informally, this just means that if a = b, then a can replace b in any mathematical expression or formula without changing its meaning.
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 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
S4 := reflexive and transitive; S5 := reflexive and Euclidean; The Euclidean property along with reflexivity yields symmetry and transitivity. (The Euclidean property can be obtained, as well, from symmetry and transitivity.) Hence if the accessibility relation R is reflexive and Euclidean, R is provably symmetric and transitive as well.