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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-
A
$2^m$
B
$2^{m^2 - m}$
C
$2^{m^2}$
D
$2^{m^2 - m} - 1$
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
$\left(^{m^{2}-m} C_{0}+^{m^{2}-m} C_{1}+\ldots . .+^{m^{2}-m} C_{m^{2}-m}\right)=2^{m^{2}-m}$
Standard 12
Mathematics
