Let $A=\{1,2,3\} .$ Then number of equivalence relations containing $(1,2)$ is
Let $A=\{1,2,3\} .$ Then number of equivalence relations containing $(1,2)$ is
It is given that $A =\{1,2,3\}$.
The smallest equivalence relation containing $(1,2)$ is given by,
$R 1=\{(1,1)\,,(2,2),\,(3,3)\,,(1,2),\,(2,1)\}$
Now, we are left with only four pairs i.e., $(2,3),\,(3,2),\,(1,3),$ and $(3,1)$
If we odd any one pair $[$ say $(2,3)]$ to $R 1,$ then for symmetry we must add $(3,2)$
Also, for transitivity we are required to add $(1,3)$ and $(3,1)$.
Hence, the only equivalence relation (bigger than $R 1$ ) is the universal relation.
This shows that the total number of equivalence relations containing $(1,2)$ is two.
The correct answer is $A$.
Similar Questions
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Let $R$ be the relation in the set $\{1,2,3,4\}$ given by $R =\{(1,2),\,(2,2),\,(1,1),\,(4,4)$ $(1,3),\,(3,3),\,(3,2)\}$. Choose the correct answer.
Let $R$ be a relation on the set of all natural numbers given by $\alpha b \Leftrightarrow \alpha$ divides $b^2$.
Which of the following properties does $R$ satisfy?
$I.$ Reflexivity $II.$ Symmetry $III.$ Transitivity
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- [KVPY 2017]
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If $R$ is a relation on the set $N$, defined by $\left\{ {\left( {x,y} \right);3x + 3y = 10} \right\}$
Statement $-1$ : $R$ is symmetric
Statement $-2$ : $R$ is reflexive
Statement $-3$ : $R$ is transitive, then thecorrect sequence of given statements is
(where $T$ means true and $F$ means false)
If $R$ is a relation on the set $N$, defined by $\left\{ {\left( {x,y} \right);3x + 3y = 10} \right\}$
Statement $-1$ : $R$ is symmetric
Statement $-2$ : $R$ is reflexive
Statement $-3$ : $R$ is transitive, then thecorrect sequence of given statements is
(where $T$ means true and $F$ means false)
The relation $R= \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)\}$ on set $A = \{1, 2, 3\}$ is
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