Power Set of Empty Set

The power set of a set is defined as the set of all possible subsets of that set. In other words, it is the collection of all possible combinations of the elements of the given set.

The empty set, also known as the null set, is a set that contains no elements. It is denoted by the symbol Ø or {}.

Calculating the Power Set

To calculate the power set of a set, we consider all possible combinations of the elements of the set.

For example, consider the set A = {1, 2}.

The power set of A is calculated as follows:
- The set A has two elements, so there are 2^2 = 4 possible combinations.
- The power set of A is {Ø, {1}, {2}, {1, 2}}.

Power Set of Empty Set

The power set of the empty set is a special case. Since the empty set contains no elements, it has no possible combinations.

Definition of Subset

A subset is a set that contains only elements that are also in another set. In other words, all the elements of the subset are also present in the larger set.

For example, if A = {1, 2, 3} and B = {2, 3}, then B is a subset of A because all the elements of B (2 and 3) are also present in A.

Number of Subsets of the Empty Set

By definition, the empty set is a subset of every set, including itself. Therefore, the power set of the empty set must include both the empty set and the set itself.

So, the power set of the empty set contains exactly two subsets: the empty set and the set itself.

Answer

Therefore, the correct answer is option A) One. The power set of the empty set has exactly one subset, which is the empty set itself.