If $A$ is an empty set, denoted as $A = \emptyset$, the power set of $A$, denoted as $P(A)$, is the set of all subsets of $A$.

Step-by-Step Explanation:

1. The empty set $A = \emptyset$ has no elements.
2. The power set $P(A)$ is the set of all subsets of $A$.
3. The empty set $\emptyset$ has exactly one subset: itself, $\emptyset$.

Thus, the power set $P(A)$ of an empty set contains only one subset, which is the empty set itself. Therefore:

$P(A) = \{ \emptyset \}$

Number of elements in $P(A)$:

• The number of elements in $P(A)$ is 1, because there is only one subset: the empty set itself.

Answer:

The number of elements in $P(A)$, where $A = \emptyset$, is $1$.

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Related Questions:

1. How do you find the power set of a set with more than one element?
2. What is the general formula for the number of elements in the power set of a set with $n$ elements?
3. How does the power set of a set with infinite elements differ from a finite set?
4. Can the power set of a set ever be infinite if the original set is finite?
5. What are some properties of power sets in set theory?

Tip: The power set $P(A)$ of any set $A$ with $n$ elements always has $2^n$ elements.