The number of relations, defined on the set { | vedclass

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(D) relation $R$ on a set $A$ with $n$ elements is reflexive if it contains all $n$ diagonal elements $(x, x)$ for all $x \in A$. For the set $A = \{a

Vedclass · Accepted Answer

$64$

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(D) relation $R$ on a set $A$ with $n$ elements is reflexive if it contains all $n$ diagonal elements $(x, x)$ for all $x \in A$. For the set $A = \{a, b, c, d\}$, $n = 4$. There are $n^2 = 16$ possible ordered pairs in $A 	imes A$. The $n = 4$ diagonal elements must be present in a reflexive relation. The remaining $n^2 - n = 16 - 4 = 12$ elements are off-diagonal pairs. For a relation to be symmetric, if $(x, y) \in R$, then $(y, x)$ must also be in $R$. These $12$ off-diagonal elements form $6$ pairs of the form ${(x, y), (y, x)}$. For each such pair, we have $2$ choices: either both are included in $R$, or neither is included. Thus, the total number of reflexive and symmetric relations is $2^6 = 64$.

The number of relations, defined on the set ${a, b, c, d}$, which are both reflexive and symmetric, is equal to:

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