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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.
In logic, equality is a primitive predicate (a statement that may have free variables) with the reflexive property (called the Law of identity), and the substitution property. From those, one can derive the rest of the properties usually needed for equality.
Reflexive and transitive: The relation ≤ on N. Or any preorder; Symmetric and transitive: The relation R on N, defined as aRb ↔ ab ≠ 0. Or any partial equivalence relation; Reflexive and symmetric: The relation R on Z, defined as aRb ↔ "a − b is divisible by at least one of 2 or 3."
[8] [9] This definition is equivalent to a partial order on a setoid, where equality is taken to be a defined equivalence relation rather than set equality. [10] Wallis defines a more general notion of a partial order relation as any homogeneous relation that is transitive and antisymmetric. This includes both reflexive and irreflexive partial ...
A relation is quasi-reflexive if, and only if, it is both left and right quasi-reflexive. The previous 6 alternatives are far from being exhaustive; e.g., the binary relation xRy defined by y = x 2 is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pair (0, 0), and (2, 4), but not (2, 2), respectively. The latter two ...
By definition, equality is an equivalence relation, meaning it is reflexive (i.e. =), symmetric (i.e. if = then =), and transitive (i.e. if = and = then =). [33] It also satisfies the important property that if two symbols are used for equal things, then one symbol can be substituted for the other in any true statement about the first and the ...
Reflexive relation, a relation where elements of a set are self-related; Reflexive user interface, an interface that permits its own command verbs and sometimes underlying code to be edited; Reflexive operator algebra, an operator algebra that has enough invariant subspaces to characterize it; Reflexive space, a subset of Banach spaces
The equality relation expressed by the sign "=" is an equivalence relation in being reflexive (everything is equal to itself), symmetric (if x is equal to y then y is equal to x) and transitive (if x is equal to y and y is equal to z then x is equal to z).