Question

If $\phi $denotes null set then $P(P(P(\phi )))$is-

Answer 1

Hint: A set which has no element is called an empty set; it is denoted by $\phi .$ The collection of all the subsets of a set is called power set. Power set of A is denoted by P(A).

Complete step-by-step solution:
Since, an empty set has no elements. We can say that the number of elements in $\phi $is zero.
Every set is a subset of itself. Hence, empty set will have one subset i.e. $\phi $
Therefore, $P(\phi ) = \left\{ \phi \right\}$
Now $P(\phi )$has one element.
Number of elements in a power set is given by formula,
$n(P(A)) = {2^{n(A)}}$
Where, $n(A)$is the number of elements in A
Therefore, number of elements in $P(P(\phi ))$will be ${2^{n(P(\phi ))}} = {2^1} = 2$
So, we can write $P(P(\phi ))$as,
$P(P(\phi )) = \left\{ {\phi ,\left\{ \phi \right\}} \right\}$
Using the same concept as above, we can observe that the number of elements in $P(P(P(\phi )))$will be
${2^{n(P(P(\phi )))}} = {2^2} = 4$
Thus, the set \[P(P(P(\phi )))\]will be written as
$P(P(P(\phi ))) = \{ \phi ,\{ \phi \} ,\{ \{ \phi \} \} ,\{ \phi ,\{ \phi \} \} \} $
Additional information:
A set containing only zero i.e. {0} is not an empty set. Because an empty set has no elements. But in this case, {0} has one element i.e. 0.
Every set has at least one subset i.e. $\phi $, hence the cardinality of a power set can never be zero.
Minimum number of elements in a power set is 1.

Notes: $\{ \phi \} $does not symbolize the empty set. It represents a set that contains an empty set as an element and hence has a cardinality one. Always make sure to use the formula $n(P(A)) = {2^{n(A)}}$ to find the number of elements in a power set so that you do not miss any element while writing the power set.