Question

If P(A) denotes the power set of A and A is the void set, then what is the number of elements in \[P\left\{ {P\left\{ {P\left\{ {P(A)} \right\}} \right\}} \right\}\] ?

Answer 1

Hint: First obtain the number of power set of A to obtain P(A). Then obtain the number of power set of P(A) to obtain P(P(A)). Similarly proceed further to obtain the required result.



Formula Used:The number of power set of a set A having n elements is \[{2^n}\] .



Complete step by step solution:The given set A is a void set,
Therefore, \[P(A) = {2^0}\]
=1
Now,
\[P\left\{ {P(A)} \right\} = {2^1}\]
=2
So,
\[P\left\{ {P\left\{ {P(A)} \right\}} \right\} = {2^2}\]
=4
Hence,
\[P\left\{ {P\left\{ {P\left\{ {P(A)} \right\}} \right\}} \right\} = {2^4}\]
=16

Therefore, \[P\left\{ {P\left\{ {P\left\{ {P(A)} \right\}} \right\}} \right\} = 16\]





Additional Information:Void set: A void set is a set that contains no element. It is also known as an empty set. It is denoted by {} or $\phi. The number of elements of a void set is 0.
Power set: All subset of a set is known as the power set of a set.
The number of elements of a power set is at least 1.
Remember $\[\phi\] is a subset of all sets.



Note: As it is given that the set A is void, sometime students write the answer as 1. But, t only P(A) is 1, here we need to calculate \[P\left\{ {P\left\{ {P\left\{ {P(A)} \right\}} \right\}} \right\}\]. Calculate the power sets one after one and obtain the required answer.