Let X be a finite set of a elements, $\mathrm{n} \in \mathrm{N}$. Then the number of un ordered pair

Question

Let X be a finite set of a elements, $\mathrm{n} \in \mathrm{N}$. Then the number of un ordered pairs $\{\mathrm{A}, \mathrm{B}\}, \mathrm{A}, \mathrm{B} \in \mathrm{P}(\mathrm{X})$ such that $\mathrm{A} \cap \mathrm{B}=\phi$ is

Professor PC Thomas Classes · Answer

The total number of ordered pairs ( $\mathrm{A}, \mathrm{B}$ ) such that $\mathrm{A} \cap \mathrm{B}=\phi$ is $3^{\mathrm{n}}$. If A or B is non empty then ( $\mathrm{A}, \mathrm{B}$ ) and ( $\mathrm{B}, \mathrm{A}$ ) are considered in $3^4$. The ordered pair ( $\phi, \phi$ ) is considered only once. So the number of subsets $\{\mathrm{A}, \mathrm{B}\}$ of $\mathrm{P}(\mathrm{X})$ with $\mathrm{A} \cap \mathrm{B}=\phi$ is $\frac{3^{\mathrm{n}}-1}{2}+1=\frac{3^{\mathrm{n}}+1}{2}$.