Question

How many elements has P(A), if $A = \phi$?

Answer 1

If we have a set ‘A’ with m elements we have the number of elements it has is $n(A) = m$. And Generally, if a set has n elements, the power set will have, ${2^n}$ elements altogether.
Thus using these results we will solve our given problem.

Complete step by step solution: We know that if A is a set with m elements i.e., $n(A) = m$,
Now, the power set of a set is the set of all subsets of A. now if a set has $A = \\{ a,b,c\\}$ the power set of the set would be, $P(A) = \\{ \phi ,\\{ a\\} ,\\{ b\\} ,\\{ c\\} ,\\{ a,b\\} ,\\{ a,c\\} ,\\{ b,c\\} ,\\{ a,b,c\\} \\}$ so, we have 8 elements in the set of P(A).
Generally, if a set has n elements, the power set will have, ${2^n}$ elements altogether.
then, $n\left[ {P\left( A \right)} \right] = {2^m}$, where m is number of elements in set A.
If $A = \phi$, then the number of elements in the set is 0, so, $n\left( A \right) = 0$
$\therefore n\left[ {P\left( A \right)} \right] = {2^0} = 1$, as ${a^0} = 1$.

Hence, P(A) has one element.

Note: In mathematics, the power set (or power set) of any set S is the set of all subsets of S, including the empty set and S itself, variously denoted as P(S). P(S), or, identifying the power set of S with the set of all functions from S to a given set of two elements, ${2^S}$. The power set is closely related to the binomial theorem. The number of subsets with k elements in the power set of a set with n elements is given by the number of combinations, $C\left( {n,\;k} \right),$ also called binomial coefficients.