This is a multiple choice question from my Text Book
Let $A=\{1,2,3\}$. The no. of relations containing $(1,2)$ and $(1,3)$ which are reflexive and Symmetric but not transitive is
(A) $1$
(B) $2$
(C) $3$
(D) $4$
My Approach:
$A=\{1,2,3\}$
Relation $R$ must contain $(1,2)$ and $(1,3)$
For $R$ to be Reflexive, it must contain $(2,2)$ and $(1,1)$
For $R$ to be symmetric, it must contain $(2,1)$ and $(3,1)$
For $R$ to not to be Transitive, it must not contain $(2,3)$ and $(3,2)$
\therefore, $R=\{(1,2),(1,3),(2,2),(1,1),(3,1),(2,1)\}$
Anyother addition to $R$ will not satisfy the stated condition.
Hence, option $A$ is correct
Am I right?
[Edit:
R contains $(3,3)$ as well]
Best Answer
You must also have $(3,3)$ for the relation to be reflexive. Because you have $(2,1)$ and $(1,3)$ you would need $(2,3)$ for the relation to be transitive. Similarly, you need $(3,2)$. If either or both of these are missing, the relation is not transitive, so there would be $3$ choices, but the requirement of symmetry says they have to both be missing and there is only one choice.