If n(A) = m , then total number of reflexive relations that can be defined on A is-

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1.Relation and Function

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If $n(A) = m$, then total number of reflexive relations that can be defined on $A$ is-

$2^m$

$2^{m^2 - m}$

$2^{m^2}$

$2^{m^2 - m} - 1$

Solution

$n(A \times A)=m^{2} .$ Now let $R$ is reflexive so $R$ must contain $\mathrm{m}$ elements i.e. $(\mathrm{i}, \mathrm{i}) \in \mathrm{R} \forall \mathrm{x} \in \mathrm{A}$

now remaining element in $(\mathrm{A} \times \mathrm{A})$ which are not in $'R'$ $=m^{2}-m$

so number of reflexive relation that can be defined on $A$ are

$\left(^{m^{2}-m} C_{0}+^{m^{2}-m} C_{1}+\ldots . .+^{m^{2}-m} C_{m^{2}-m}\right)=2^{m^{2}-m}$

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