Question

Let the number of elements in a set ${}^{\prime }{A}^{\prime }$ be ${}^{\prime }{n}^{\prime }.$ A set ${}^{\prime }{C}^{\prime }$ is defines such that the $C=\left\{\right(x,y)|x,y\in P(A)\text{and}x\cap y=\varphi \},$ where $P\left(A\right)$ is power set of ${}^{\prime }{A}^{\prime },$then cardinal number of ${}^{\prime }{C}^{\prime }$ is

• A. $2^{2n}$
• B. $3^n$
• C. ${2.3}^n$
• D. none of these

Answer 1

The correct option is B $3^n$


From the above table $x\cap y=\varphi$ is possible in last three cases. Therefore every element has 3 possibilities.
So, cardinal number of $C=n\left(C\right)=3^n$ (Empty set is allowed)