1.Relation and FunctionMedium

ગણ $A=\{1,2,3\}$ લો. $(1, 2)$ ને સમાવતા સામ્ય સંબંધોની સંખ્યા ........... છે.

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 $A$.

ધારોકે $R$ પરના બે સંબંધો $R_{1}$ અને $R_{2}$ નીયે મુજબ વ્યાખ્યાયિત છે: $a R_{1} b \Leftrightarrow a b \geq 0$ અને $a R_{2} b \Leftrightarrow a \geq b$, તો

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