The Power Set — Mathematics Knowledge System

Overview

The power set of a set A, written P(A) or 2^A, is the set of all subsets of A, including the empty set and A itself. If A has n elements, P(A) has exactly 2^n elements, since each element independently is either included or excluded from a given subset. The power set is itself a set — its elements are sets — which makes it a natural way to build richer structures (topologies, sigma-algebras, Boolean algebras) on top of a base set.

Intuition

Think of building a subset of A by walking through each element one at a time and flipping a coin: heads, include it; tails, leave it out. With n elements there are n independent coin flips, so there are 2^n possible outcomes — 2^n distinct subsets. This binary-choice picture is why power sets are so tightly linked to binary strings: a subset of {1,...,n} corresponds exactly to a length-n string of 0s and 1s recording which elements were kept.

Formal Definition

Definition

For a set A, the power set collects every subset of A:

\mathcal{P}(A) = \{S : S \subseteq A\}

Definition of the power set

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

Cardinality of the power set (finite A)

\emptyset \in \mathcal{P}(A) \quad \text{and} \quad A \in \mathcal{P}(A)

The empty set and A itself are always members

|A| < |\mathcal{P}(A)|

Cantor's theorem: the power set is always strictly bigger, even for infinite A

Worked Examples

A 2-element set has 2² = 4 subsets: the empty set, each singleton, and the whole set.
$\mathcal{P}(\{a,b\}) = \{\emptyset, \{a\}, \{b\}, \{a,b\}\}$

Answer: P({a,b}) = {∅, {a}, {b}, {a,b}}, with 4 elements.

Practice Problems

Difficulty 2/10

How many elements does the power set of a 5-element set have?

Difficulty 3/10

Which of the following is always an element of P(A), for any set A?

Only ∅Only ABoth ∅ and ANeither, unless stated

Difficulty 6/10

Explain briefly why no set A can be put in bijection with its own power set P(A).

Quiz

Summary

P(A), the power set of A, is the set of all subsets of A, including ∅ and A itself.
For a finite set with n elements, |P(A)| = 2^n.
Cantor's theorem: |A| < |P(A)| for every set A, finite or infinite, which generates an endless hierarchy of larger infinities.

References

WebsiteWikipedia — Power set