Power Set¶

Definition¶

The power set of a set $A$, denoted $2^A$ or $\mathcal{P}(A)$, is the set of all subsets of $A$:

Interpretation¶

The power set collects every possible choice of elements from $A$, including the empty choice and the full set itself. Its elements are themselves sets: for $A=\{a,b,c\}$ one writes $a\in A$$A$, but $\{a\}\in 2^A$$2^A$.

The figure shows the eight subsets of $\{a,b,c\}$: $\varnothing$ at the bottom, then the three one-element and the three two-element subsets, and $\{a,b,c\}$ at the top. A line joins two subsets that differ by one element. The count $2^3=8$ comes from deciding for each of the three elements independently whether it is in or out, which also explains the notation $2^A$.

The same object appears in probability: every event of a die roll is a subset of $S=\{1,\ldots,6\}$$\{1,\dots ,6\}$, so it is an element of $2^S$, which has $2^6=64$ elements.

Key Results¶

• If $A$ is finite with $n$ elements, then $2^A$ has $2^n$ elements.
• Cantor's theorem shows that for any set $A$, the power set $2^A$ has strictly larger cardinality than $A$.

Examples¶

• If $A=\{a,b\}$$\{a,b\}$, then $2^A=\{\varnothing,\{a\},\{b\},\{a,b\}\}$$\{\varnothing ,\{a\},\{b\},\{a,b\}\}$.
• If $A=\varnothing$$\varnothing$, then $2^A=\{\varnothing\}$$\{\varnothing \}$, so the power set of the empty set has one element.

Prerequisites¶

• Set
• Subset

Related Concepts¶

• Continuum Hypothesis
• Sample Space
• Aleph Null

Sources¶

• Math II: Linear Algebra · Week 1

Aliases: powerset Keywords: set of all subsets Symbols: element of, subset

Last updated: 20 October 2026 03:02