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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 relation that is reflexive, antisymmetric, and transitive. Strict partial order A relation that is irreflexive, asymmetric, and transitive. Total order A relation that is reflexive, antisymmetric, transitive and connected. [20] Strict total order A relation that is irreflexive, asymmetric, transitive and connected. Uniqueness properties: One ...
The unit sphere can be replaced with the closed unit ball in the definition. Namely, a normed vector space is uniformly convex if and only if for every < there is some > so that, for any two vectors and in the closed unit ball (i.e. ‖ ‖ and ‖ ‖) with ‖ ‖, one has ‖ + ‖ (note that, given , the corresponding value of could be smaller than the one provided by the original weaker ...
A relation is called reflexive if it relates every element of to itself. For example, if X {\displaystyle X} is a set of distinct numbers and x R y {\displaystyle xRy} means " x {\displaystyle x} is less than y {\displaystyle y} ", then the reflexive closure of R {\displaystyle R} is the relation " x {\displaystyle x} is less than or equal to y ...
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 resulting relation is reflexive since the preorder is reflexive; transitive by applying the transitivity of twice; and symmetric by definition. Using this relation, it is possible to construct a partial order on the quotient set of the equivalence, S / ∼ , {\displaystyle S/\sim ,} which is the set of all equivalence classes of ∼ ...
In universal algebra and lattice theory, a tolerance relation on an algebraic structure is a reflexive symmetric relation that is compatible with all operations of the structure. Thus a tolerance is like a congruence , except that the assumption of transitivity is dropped. [ 1 ]
In the area of mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) X such that the canonical evaluation map from X into its bidual (which is the strong dual of X) is bijective.