Q.16 Let $A = \{1, 2, 3\}$. Then number of relations containing $(1, 2)$ and $(1, 3)$ which are reflexive and symmetric but not transitive is(A) 1 (B) 2 (C) 3 (D) 4

Answers (1)

$A = \{1, 2, 3\}$

The smallest  relations containing $(1, 2)$ and $(1, 3)$ which are
reflexive and symmetric but not transitive is given by

$R = \left \{ (1,1),(2,2),(3,3),(1,2),(1,3),(2,1),(3,1) \right \}$

$(1,1),(2,2),(3,3) \in R$  , so relation R is reflexive.

$(1,2),(2,1) \in R$    and   $(1,3),(3,1) \in R$  , so relation R is symmetric.

$(2,1),(1,3) \in R$  but $(2,3) \notin R$  , so realation R is not transitive.

Now, if we add any two pairs $(2,3)$   and $(3,2)$ to relation R, then relation R  will become transitive.

Hence, the total number of the desired relation is one.

Thus, option A is correct.

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