Question

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

Answer 1

$A=\{1,2,3\}$
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 $A$.