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Q.
If the number of reflexive relations defined on a set $A$ is 64 , then the number of elements in $A$ is
Sets and Relations
Solution:
We have number of reflexive relations defined on $A$ is $2^{n^2-n}$, where $n$ is the cardinal number of $A$.
Given, $2^{n^2-n}=64 \Rightarrow 2^{n^2-n}=2^6$
$\Rightarrow n^2-n=6 \Rightarrow n(n-1)=3 \times 2$
$\therefore n=3 .$