$$ (A) \quad 1\qquad(B) \quad 2 \qquad(C) \quad 3 \qquad(D)\quad 4\qquad $$

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- A relation R in a set A is called reflexive. if $(a,a) \in R\;for\; all\; a\in A$
- A relation R in a set A is called symmetric. if $(a_1,a_2) \in R\;\Rightarrow \; (a_2,a_1)\in R \;$ for $\;a_1,a_2 \in A$
- A relation R in a set A is called transitive. if $(a_1,a_2) \in\; R$ and $(a_2,a_3)\in R \Rightarrow \;(a_1,a_3)\in R\; $for all $\; a_1,a_2,a_3 \in A$

Consider the relation $R$ in $A=\{1,2,3\}\;$ where $R=\{(1,1),(2,2),(3,3)(1,2)(1,3)(2,1)(3,1)(3,2)(2,3)\}$

We need to work with the relations that contains $(1,2), (3,1)$

Relation R is reflexive since $(1,1)(2,2)(3,3) \in R$

Relation R is symmetric since $(1,2),(2,1) \in R (1,3)(3,1) \in R$

Relation R is not transitive since $(3,1)(1,2) \in R\;but \;(3,2) \not \in R$

Therefore the total number of relation containing (1,2)(1,3) which are reflexive ,symmetric but not transitive in 1

However if we add the pair (3,2) and (2,3) to relation R then it will become transitive.

Therefore, the correct answer is 1 (A).

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