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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.
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 ...
The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. A simpler example is equality. Any number a {\displaystyle a} is equal to itself (reflexive).
The promised geometric property of reflexive Banach spaces is the following: if is a closed non-empty convex subset of the reflexive space , then for every there exists a such that ‖ ‖ minimizes the distance between and points of .
Euclidean geometry is an example of synthetic geometry, ... Things that coincide with one another are equal to one another (reflexive property).
A reflexive, weak, [1] or non-strict partial order, [2] commonly referred to simply as a partial order, is a homogeneous relation ≤ on a set that is reflexive, antisymmetric, and transitive. That is, for all a , b , c ∈ P , {\displaystyle a,b,c\in P,} it must satisfy:
For example, "is a blood relative of" is a symmetric relation, because x is a blood relative of y if and only if y is a blood relative of x. Antisymmetric for all x, y ∈ X, if xRy and yRx then x = y. For example, ≥ is an antisymmetric relation; so is >, but vacuously (the condition in the definition is always false). [11] Asymmetric
A reflection through an axis. In mathematics, a reflection (also spelled reflexion) [1] is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as the set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.