In Maths, a binary relation R across a set X is reflexive if each element of set X is related or linked to itself. In terms of relations, this can be defined as (a, a) ∈ R ∀ a ∈ X or as I ⊆ R where I is the identity relation on A. Thus, it has a reflexive property and is said to hold reflexivity.

They can act as either objects or indirect objects. The nine English reflexive pronouns are myself, yourself, himself, herself, oneself, itself, ourselves, yourselves, and themselves.

A relation R in a set A is not reflexive if there be at least one element a ∈ A such that (a, a) ∉ R. Consider, for example, a set A = {p, q, r, s}. The relation R1 = {(p, p), (p, r), (q, q), (r, r), (r, s), (s, s)} in A is reflexive, since every element in A is R1-related to itself.

Definition 1. An equivalence relation is a relationship on a set, generally denoted by “∼”, that is reflexive, symmetric, and transitive for everything in the set. Example: The relation “is equal to”, denoted “=”, is an equivalence relation on the set of real numbers since for any x, y, z ∈ R: 1.

The smallest reflexive relation on set {1,2,3,4} is {(1,1),(2,2),(3,3),(4,4)}.

For any set S the smallest equivalence relation is the one that contains all the pairs (s,s) for s∈S. It has to have those to be reflexive, and any other equivalence relation must have those. The largest equivalence relation is the set of all pairs (s,t).

Show that the given relation R is an equivalence relation, which is defined by (p, q) R (r, s) ⇒ (p+s)=(q+r) Check the reflexive, symmetric and transitive property of the relation x R y, if and only if y is divisible by x, where x, y ∈ N.

There are five distinct equivalence classes, modulo 5: [0], [1], [2], [3], and [4]. {x ∈ Z | x = 5k, for some integers k}. Definition 5. Suppose R is an equivalence relation on a set A and S is an equivalence class of R.

two possible relations