The number of reflexive relations on a set A | vedclass

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(D) relation $R$ on a set $A$ is reflexive if $(a, a) \in R$ for all $a \in A$. There are $n$ such elements of the form $(a, a)$ which must be present

Vedclass · Accepted Answer

$2^{n^2-n}$

Vedclass · Answer

(D) relation $R$ on a set $A$ is reflexive if $(a, a) \in R$ for all $a \in A$. There are $n$ such elements of the form $(a, a)$ which must be present in the relation. The total number of possible ordered pairs in the Cartesian product $A 	imes A$ is $n^2$. Since the $n$ diagonal elements $(a, a)$ are fixed (must be included), we have $n^2 - n$ remaining pairs to choose from. Each of these $n^2 - n$ pairs can either be included or excluded from the relation, giving $2$ choices for each. Therefore, the total number of reflexive relations is $2^{n^2 - n}$.

The number of reflexive relations on a set $A$ of $n$ elements is equal to

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