Question

Let $P\left(A\right)$ represents the power set of the $A$ and $n\left(A\right)$ represents the cardinal number of $A$. If $n\left(P\right(P\left(B\right)\left)\right)+n\left(P\right(P\left(A\right)\left)\right)=20$ then $n\left(A\right)+n\left(B\right)$ is equal to

• A. $6$
• B. $8$
• C. $4$
• D. $3$

Answer 1

Answer: (D)

$3$

Let,

Cardinal no. in A and B are m and n respectively.

then,

$n\left(A\right)=m,n\left(B\right)=n$

As we know that,

$n\left(P\right(A\left)\right)=2^m$

$n\left(P\right(P\left(A\right)\left)\right)=2^{2^m}$

$n\left(P\right(B\left)\right)=2^n$

$n\left(P\right(P\left(B\right)\left)\right)=2^{2^n}$

then,

$n\left(P\right(P\left(A\right)\left)\right)+n\left(P\right(P\left(B\right)\left)\right)=20$

$2^{2^m}+2^{2^n}=20=16+4=2^4+2^2$

$2^{2^m}+2^{2^n}=2^{2^2}+2^{2^1}$

by comparing we get

$m=2,n=1$