Question

Define a reflexive relation.

Answer 1

Hint: Relations are related to the cartesian product of the sets or bunch of points.In math, there are nine kinds of relations which are empty relation, full relation, reflexive relation, irreflexive relation, symmetric relation. Further, there is antisymmetric relation, transitive relation, equivalence relation, and finally asymmetric relation.A relation is a reflexive relation If every element of set A maps to itself. I.e for every a ∈ A,(a, a) ∈ R.

Complete step-by-step answer:
A relation is a reflexive relation If every element of set A maps to itself. I.e for every a ∈ A,(a, a) ∈ R.
OR
A relation $R$ from a non-empty set $A$ to a non-empty set $B$ is a subset of the cartesian product $A \times B$.
It maps elements of one set to another set. The subset is derived by describing a relationship between the first element and the second element of the ordered pair $A \times B$.

Domain: The set of all first elements of the ordered pairs in a relation $R$ from a set $A$ to a set $B$ is called the domain of the relation $R$.All the elements of set $A$ is called domain of $R$.
Codomain: All the elements of set $B$ are called codomain of $R$.
Range: All of the values that come out of a relation are called the range. Range may also be referred to as "image".: The set of all second elements of the ordered pairs in a relation $R$ from a set $A$ to a set $B$ can be referred to as range of $R$.

Reflexive Relation
A relation R in A is said to be reflexive if $a{\text{ R }}a$ for all $a \in A$.
It can also be stated as a relation in a set $A$ is called reflexive relation if $\left( {a,a} \right) \in {\text{R}}$ for every element $a \in A$.
For example,
Let \[A{\text{ }} = {\text{ }}\left\{ {0,{\text{ }}1,{\text{ }}2,{\text{ }}3} \right\}\] and define a relation R on $A$ as follows: \[R{\text{ }} = {\text{ }}\left\{ {\left( {0,{\text{ }}0} \right),{\text{ }}\left( {0,{\text{ }}1} \right),{\text{ }}\left( {0,{\text{ }}3} \right),{\text{ }}\left( {1,{\text{ }}0} \right),{\text{ }}\left( {1,{\text{ }}1} \right),{\text{ }}\left( {2,{\text{ }}2} \right),{\text{ }}\left( {3,{\text{ }}0} \right),{\text{ }}\left( {3,{\text{ }}3} \right)} \right\}\]
From $R$ above, it is clear that
$
  \left( {0,0} \right) \in {\text{R}} \\
  \left( {1,1} \right) \in {\text{R}} \\
  \left( {2,2} \right) \in {\text{R}} \\
  \left( {3,3} \right) \in {\text{R}} \\
$
Since for every element in \[A{\text{ }} = {\text{ }}\left\{ {0,{\text{ }}1,{\text{ }}2,{\text{ }}3} \right\}\], there exists a ordered pair $\left( {a,a} \right) \in {\text{R}}$,hence $R$ is reflexive in set $A$.

Note: Relations are one of the means of joining sets or subsets of the cartesian product. Relations and functions are different from each other. Any relation which is reflexive, symmetric and transitive is called an equivalence relation.A function is a relation which describes that there should be only one output for each input.