Set Powers

Why

We want to consider all the subsets of a given set.

Definition

We do not yet have a principle stating that such a set exists, but our intuition suggests that it does.

We call the existence of this set the principle of powers and we call the set the power set.¹ As usual, the principle of extension gives uniqueness (see Set Equality). The power set of a set includes the set itself and the empty set.

Notation

Let $A$ denote a set. We denote the power set of $A$ by $\powerset{A}$, read aloud as “powerset of A.” $A \in \powerset{A}$ and $\varnothing \in \powerset{A}$. However, $A \subset \powerset{A}$ is false.

Examples

Let $a, b, c$ denote distinct objects. Let $A = \set{a, b ,c}$ and $B = \set{a, b}$. Then $B \subset A$. In other notation, $B \in \powerset{A}$. Showing each of the following is straightforward.

1. The empty set: $\powerset{\varnothing} = \set{\varnothing}$
2. Singletons: $\powerset{\set{a}} = \set{\varnothing, \set{a}}$
3. Pairs: $\powerset{\set{a, b}} = \set{\varnothing, \set{a}, \set{b}, \set{a, b}}$
4. Triples:

   \[ \powerset{\set{a, b, c}} = \{ \varnothing, \set{a}, \set{b}, \set{c}, \set{a, b}, \set{b, c}, \set{a, c}, \set{a, b, c} \} \]

Properties

We can guess the following easy properties.²

We call $A$ and $\varnothing$ the improper subsets of $A$. All other subsets we call proper.

Basic fact

1. This terminology is standard, but unfortunate. Future editions may change these terms. ↩︎
2. Future editions will expand this account. ↩︎