Power Sets

What is a Power Set?

You have a set. What are all the possible subsets you could make from it?

Collect them all into one set — that’s the power set.

Symbol: $\mathcal{P}(A)$ or $2^A$

Example

$A = \{1, 2\}$

List all subsets of A:

• $\emptyset$ (empty set — always a subset)
• $\{1\}$
• $\{2\}$
• $\{1, 2\}$ (the set itself — always a subset)

The power set collects them all:

$\mathcal{P}(A) = \{\emptyset, \{1\}, \{2\}, \{1, 2\}\}$

The power set is a set of sets.

How Many Subsets?

For a set with $n$ elements:

$|\mathcal{P}(A)| = 2^n$

Why? For each element, you have 2 choices: include it or don’t.

This is why the notation $2^A$ makes sense — the size is $2^n$.

Larger Example

$A = \{a, b, c\}$

All 8 subsets:

$\mathcal{P}(A) = \{\emptyset, \{a\}, \{b\}, \{c\}, \{a,b\}, \{a,c\}, \{b,c\}, \{a,b,c\}\}$

Special Cases

Power set of empty set:

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

The empty set has one subset: itself. That’s $2^0 = 1$.

Power set of a singleton:

$\mathcal{P}(\{x\}) = \{\emptyset, \{x\}\}$

Two subsets: empty set and the set itself.

Key Facts

Every power set contains:

• $\emptyset$ (the empty set is a subset of everything)
• $A$ itself (every set is a subset of itself)

So for any non-empty set A:

$\emptyset \in \mathcal{P}(A)$

The empty set is a subset of A, so it’s in the power set.

$A \in \mathcal{P}(A)$

A is a subset of itself, so it’s in the power set.

The Formal Definition

$\mathcal{P}(A) = \{S \mid S \subseteq A\}$

This reads: “The set of all S such that S is a subset of A.”

Power set = all possible subsets collected into one set.