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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).
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:
If the normed space X is complete and satisfies the slightly stronger property of being uniformly convex (which implies strict convexity), then it is also reflexive by Milman–Pettis theorem. Properties
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 .
For example: "An even number is an integer which is divisible by 2." An extensional definition instead lists all objects where the term applies. For example: "An even number is any one of the following integers: 0, 2, 4, 6, 8..., -2, -4, -8..." In logic, the extension of a predicate is the set of all things for which the predicate is true. [48]