# 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.

For every set, there exists a set of its
subsets.

We call the existence of this set the
principle of powers and
we call the set the power
set.
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.

- The empty set: $\powerset{\varnothing} = \set{\varnothing}$
- Singletons: $\powerset{\set{a}} = \set{\varnothing, \set{a}}$
- Pairs: $\powerset{\set{a, b}} = \set{\varnothing, \set{a},
\set{b}, \set{a, b}}$
- 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.

$\varnothing \in \powerset{A}$

$A \in \powerset{A}$

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

# Basic fact

$E \subset F \implies \powerset{E} \subset
\powerset{F}$