The power set of the set { φ } is

Concept

For any set S, the power set P(S) is the set of all subsets of S, including the empty set and S itself. If S has n elements, then P(S) has 2n elements. In this question the symbol φ denotes the empty set.

Application

The given set is { φ }. It has one element, namely φ, so n = 1 and its power set must have 21 = 2 elements. Listing every subset of { φ }:

1. The subset that takes none of the elements of { φ } is the empty set, φ.

2. The subset that takes the one available element φ is { φ }.

3. Collecting both subsets gives P({ φ }) = { φ, { φ } }.

Cross-check

Two subsets were listed and 21 = 2 were expected, so the count agrees. The distinction that carries the whole question is that φ is the empty set, a set holding nothing, while { φ } is a set holding one thing — that empty set. They are different objects, so both appear as separate members of the power set.

• { φ } has one member, so it is the power set of φ itself, because 20 = 1.

• { 0 } has one member, the number 0, which is not a subset of { φ }.

• { 0, φ, { φ } } has three members, and 3 is not a power of 2, so it is the power set of no set at all.