Power Set — Definition, Cardinality, and Examples

TL;DR

The power set of a set AAA, written P(A), is the set of all subsets of AAA, including the empty set ∅ and AAA itself. This article defines P(A), proves the cardinality rule ∣P(A)∣=2^n for a set of n elements, lists the properties, and works through examples from the empty set up to four-element sets.

What Is the Cardinality of a Power Set?

For a finite set A with n elements, the power set has exactly 2^n elements:

∣P(A)∣=2^{n},where n=∣A∣.

The number of elements in a set is its cardinality, so this rule says the power set's cardinality is two raised to the original set's cardinality.

Why 2^n? (the one-line proof)

To build any subset of AAA, you walk through the n elements one at a time and make a single binary decision for each: in or out. Two choices, made independently, n times over:

2×2×⋯×2⏟n times=2^{n}.

Every distinct sequence of in/out choices produces a distinct subset, and every subset corresponds to exactly one such sequence, so the count of subsets is exactly 2^n. That correspondence is the proof; the formula is just its arithmetic.

Properties of the Power Set

• The empty set is always in P(A). ∅⊆A for every A, so ∅ is always a member of the power set.

• AAA itself is always in P(A). A set is a subset of itself, so A ∈ P(A).

• ∣P(A)∣=2^n is always even for n≥1, and never zero, because even an empty set has a power set.

• Bigger sets have strictly bigger power sets. ∣P(A)∣=2^n>n for every n, so a set can never be the same size as its power set. (This is the finite shadow of Cantor's theorem, which extends the result to infinite sets.)

• Power sets exist for infinite sets too; the power set of ℕ is infinite and provably "larger" than ℕ itself.

Examples of Power Set

Example 1

Find the power set of the empty set ∅.

The empty set has n=0 elements.

By the rule, ∣P(∅)∣=2^0=1.

Its only subset is ∅ itself.

P(∅)=∅.

Final answer: P(∅)=∅, a singleton whose one element is the empty set.

Example 2

A student writes the power set of A={a}.

This is the most common first attempt, so it is worth seeing why it breaks. The instinct is to "wrap the elements in braces" and stop.

Check it against the formula. A has n=1 element, so ∣P(A)∣ should be 2^1=2, but {a} has only one member, and that member is a, an object, not a set. The elements of a power set must be subsets.

The two subsets of {a} are ∅ and {a}.

P(A)=∅,a.

Final answer: P(a)=∅,a, with 2 elements. The empty set is the subset students drop most often.

Example 3

Write the power set of A={1, 2, 3} and confirm its size.

n=3, so expect ∣P(A)∣=2^3=8.

List subsets by size:

• size 0: ∅
• size 1: {1}, {2}, {3}
• size 2: {1, 2}, {1, 3}, {2, 3}
• size 3: {1, 2, 3}

P(A)=∅,1,2,3,{1,2},{1,3},{2,3},{1,2,3}.

Count: 1+3+3+1=8. ✓

Final answer: 8 subsets, matching 2^3.

Example 4

A set has 64 subsets. How many elements does it have?

Set 2^n=64.

Since 64=2^6, we get n=6.

Final answer: the set has 6 elements.

Example 5

Find the power set of B={x, y} and identify which elements of P(B) are themselves singletons.

n=2, so ∣P(B)∣=2^2=4.

P(B)=∅,x,y,{x,y}.

The singleton members are {x} and {y}.

Final answer: P(B)=∅,x,y,{x,y}; the singletons inside it are {x} and {y}.

Example 6

For A={1, 2, 3, 4}, how many subsets contain the element 1?

Total subsets: 2^4=16.

Fix 1 as "in." The remaining three elements (2,3,4) are each independently in or out: 2^3=8 ways.

So 8 of the 16 subsets contain 1.

Final answer: 8 subsets contain the element 1.

Why Counting Subsets Matters

"Why would anyone want every subset of a set?" Because a power set is the mathematical model of every possible combination of choices, and that shape appears everywhere once you look.

• Every yes/no configuration is a subset. Toppings on a pizza, switches in a circuit, features turned on in a settings menu.

• It is the engine behind truth tables. n logical variables have 2^n rows in their truth table.

• It is how mathematics discovered different sizes of infinity. Cantor proved that P(A) is always strictly larger than A.

Where the Power Set Trips Students Up

Mistake 1: Forgetting ∅ and A itself

Don't do this: list only the "real" subsets, the singletons and pairs, and leave out the empty set or the full set.

The correct way: the empty set and the original set are always subsets.

Mistake 2: Writing elements instead of subsets

Don't do this: write P(a,b)={a,b}.

The correct way: the elements of a power set are sets, so each must wear its own braces.

Mistake 3: Using n^2 or 2n instead of 2^n

Don't do this: misremember the formula.

The correct way: it is 2^n (two to the power of n).

Conclusion

• The power set P(A) is the set of all subsets of A, including ∅ and A itself.
• Its cardinality is ∣P(A)∣=2^n, where n is the number of elements in A.
• The 2^n rule comes from an independent in/out choice for each of the n elements.
• Elements of a power set are sets, so each is written in its own braces.
• Power sets grow explosively, which is why they model "all combinations" and why brute-force subset search is expensive.