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1.Relation and Function
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यदि $A =\{1,2,3\}$ हो तो अवयव $(1,2)$ वाले तुल्यता संबंधों की संख्या है।
A
$2$
B
$1$
C
$3$
D
$4$
Solution
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 $A$.
Standard 12
Mathematics
