Question

Give an example of a relation which is symmetry only.

Answer 1

Hint- In order to deal with this question we will first assume a set $A$ and find the relation \[R\] on set $A$ further we will check each condition of reflexive, symmetry and transitive and according to it we will comment that our set is symmetry only.

Complete step-by-step answer:
Let $A = \{ 1,2,3\} $
Let relation R on set A be
Let $R = \{ (1,2),(2,1)\} $
Now we will check the condition of reflexive
So as we know that if the relation is reflexive then $(a,a) \in R$ for every $a \in \{ 1,2,3\} $
Since, $(1,1),(2,2),(3,3)$ doesn’t belongs to $R$
$R = \{ (1,2),(2,1)\} $ is not reflexive
Now we will check the condition of symmetry
So as we know that if the relation is symmetry if $(a,b) \in R,$then $(b,a) \in R$
Since, $(1,2) \in R,(2,1) \in R$
$R = \{ (1,2),(2,1)\} $ is symmetry
At last we will check the condition of transitive
So as we know that to check whether transitive or not
If $(a,b) \in R\& (b,a) \in R,$then $(a,c) \in R$
If \[a{\text{ }} = {\text{ }}1,{\text{ }}b{\text{ }} = {\text{ }}2\] but there is no \[c\] ( no third element)
Similarly, If \[a{\text{ }} = {\text{ }}2,{\text{ }}b{\text{ }} = {\text{ }}1\]but there is no \[c\] ( no third element)
Here, $R = \{ (1,2),(2,1)\} $ is not transitive
Hence, relation $R = \{ (1,2),(2,1)\} $ is symmetry only and it is not reflexive and transitive.

Note- For a relation \[R\] in set $A$. Relation is reflexive if $(a,a) \in R$ for every $a \in A$, It would be symmetric if $(a,b) \in R$, then $(b,a) \in R$ and transitive if $(a,b) \in R\& (b,c) \in R,$ then $(a,c) \in R$. If a relation is reflexive, symmetric and transitive then it is an equivalence relation.