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Q.
Let $A =\{1,2,3\} .$ Then, the number of relations containing $(1,2)$ and $(1,3)$, which are reflexive and symmetric but not transitive, is
Relations and Functions - Part 2
Solution:
Let $R$ be a relation containing $(1,2)$ and $(1,3)$ R is reflexive,
if $(1,1),(2,2),(3,3) \in R$
Relation $R$ is symmetric, if $(2,1) \in R$ but $(3,1) \not \in R$.
But relation $R$ is not transitive as $(3,1),(1,2) \in R$ but $(3,2) \notin R$
Now, if we add the pair $(3,2)$ and $(2,3)$ to relation $R$,
then relation $R$ will become transitive.
Hence, the total number of desired relations is one.