Let $A=\{1,2,3\} .$ Then number of relations containing $(1,2)$ and $(1,3)$ which are reflexive and symmetric but not transitive is
Let $A=\{1,2,3\} .$ Then number of relations containing $(1,2)$ and $(1,3)$ which are reflexive and symmetric but not transitive is
The given set is $A=\{1,2,3\}$.
The smallest relation containing $(1,2)$ and $(1,3)$ which is reflexive and symmetric, but not transitive is given by:
$R=\{(1,1),\,(2,2),\,(3,3),\,(1,2),\,(1,3),\,(2,1),\,(3,1)\}$
This is because relation $R$ is reflexive as $(1,1),\,(2,2),\,(3,3) \in R$
Relation $R$ is symmetric since $(1,2),\,(2,1) \in R$ and $(1,3),\,(3,1) \in R$
But relation $R$ is not transitive as $(3,1),\,(1,2) \in R,$ but $(3,2)\notin R$
Now, if we add any two pairs $(3,2)$ and $(2,3) $ (or both) to relation $R$, then relation $R$ will become transitive.
Hence, the total number of desired relations is one.
The correct answer is $D$.
Similar Questions
The probability that a relation $R$ from $\{ x , y \}$ to $\{ x , y \}$ is both symmetric and transitive, is equal to
The probability that a relation $R$ from $\{ x , y \}$ to $\{ x , y \}$ is both symmetric and transitive, is equal to
- [JEE MAIN 2022]
Let $A = \{1, 2, 3, 4\}$ and $R$ be a relation in $A$ given by $R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (3, 1), (1, 3)\}$. Then $R$ is
Let $A = \{1, 2, 3, 4\}$ and $R$ be a relation in $A$ given by $R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (3, 1), (1, 3)\}$. Then $R$ is
The relation $R =\{( a , b ): \operatorname{gcd}( a , b )=1,2 a \neq b , a , b \in Z \}$ is:
The relation $R =\{( a , b ): \operatorname{gcd}( a , b )=1,2 a \neq b , a , b \in Z \}$ is:
- [JEE MAIN 2023]
If $R$ is an equivalence relation on a set $A$, then ${R^{ - 1}}$ is
If $R$ is an equivalence relation on a set $A$, then ${R^{ - 1}}$ is
Show that the relation $\mathrm{R}$ in the set $\mathrm{Z}$ of integers given by $\mathrm{R} =\{(\mathrm{a}, \mathrm{b}): 2$ divides $\mathrm{a}-\mathrm{b}\}$ is an equivalence relation.
Show that the relation $\mathrm{R}$ in the set $\mathrm{Z}$ of integers given by $\mathrm{R} =\{(\mathrm{a}, \mathrm{b}): 2$ divides $\mathrm{a}-\mathrm{b}\}$ is an equivalence relation.