If $A = {1, 2, 3},$ then number of reflexive relations that can be defined on $A$ is

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Solution

$A = {1, 2, 3}$
We know that for a reflexive relation $(a, a) \in R \forall a \in A$
So, these three $(1, 1), (2, 2), (3, 3)$ elements should always be present for relation to be reflexive.
Now, total number elements in $n (A \times A) = 9$
Total ordered pairs $= 9$
Among them $3$ should be there for reflexive relation.
From remaining $6$ elements, No of ways in which elements can be selected $=$ all possible selections from $6$ distinct elements
$=^{6} C_{0} +^{6} C_{1} + . . . +^{6} C_{6} = 2^{6} = 64$
So, $64$ reflexive relations can be defined on $A$ .

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