ALGEBRAAlgebraMathematics Calculator
𝒫

Power Set

P(S) = set of all subsets of S. |P(S)| = 2^n for n elements. Each element: in or out → 2 choices. Proper subsets exclude S itself. k-subsets: C(n,k) = n!/(k!(n−k)!).

Concept Fundamentals
|P(S)|=2ⁿ
Cardinality
2ⁿ−1
Proper
C(n,k)
k-subsets
In/out per element
Binary

Did our AI summary help? Let us know.

Each element: in or out — 2 choices. n elements → 2^n subsets. Proper subsets exclude S. Count: 2^n − 1. C(n,k) = number of ways to choose k elements from n.

Key quantities
|P(S)|=2ⁿ
Cardinality
Key relation
2ⁿ−1
Proper
Key relation
C(n,k)
k-subsets
Key relation
In/out per element
Binary
Key relation

Ready to run the numbers?

Why: Power sets model all possible combinations — used in logic, topology, and computer science. |P(S)|=2^n grows exponentially. k-subsets count combinations without repetition.

How: List elements. For each of 2^n binary strings (0=out, 1=in), form subset. Proper: exclude full set. k-subsets: C(n,k)=n!/(k!(n−k)!) = binomial coefficient.

Each element: in or out — 2 choices. n elements → 2^n subsets.Proper subsets exclude S. Count: 2^n − 1.
Sources:Khan Academy — Set TheoryWolfram MathWorld — Power Set

Run the calculator when you are ready.

Generate Power SetAll subsets, k-subsets

Enter Set

power_set.sh
CALCULATED
$ powerset --set=a, b, c
S
{a, b, c}
|P(S)|
8
2^3
Proper subsets
7
Power set (first 8)
∅
{a}
{b}
{a, b}
{c}
{a, c}
{b, c}
{a, b, c}
Power Set Calculator
|P({a, b, c})| = 2^3 = 8
Proper subsets: 7
numbervibe.com
Share:

Subsets by Size

Size Distribution

📐 Step-by-Step Breakdown

INPUTS
Original set S
{a, b, c}
|S|
3
METHOD
|P(S)| = 2^n
2^3 = 8
RESULT
Proper subsets
7

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📐

S={a,b} → P(S)={∅,{a},{b},{a,b}}. |P(S)|=4=2².

— Example

🔢

C(5,2)=10 — ways to choose 2 from 5 elements.

— Combination

📋 Key Takeaways

  • • Power set P(S) = set of all subsets of S, including ∅ and S itself.
  • • |P(S)| = 2^n — each element is either in or out; 2 choices per element.
  • • Binary representation: Integer 0..2^n−1 encodes each subset (bit j = 1 ⇒ element j included).
  • • Cantor's theorem: |P(S)| > |S| for any set S. No set equals its power set.
  • • k-element subsets: C(n,k). Sum: Σ C(n,k) = 2^n.

💡 Did You Know?

🔢P({a,b}) = {∅, {a}, {b}, {a,b}}. Four subsets from two elements.Source: Definition
💻Subset sum, knapsack, and many CS problems enumerate power set (2^n complexity).Source: Computer Science
📐Boolean algebra: power set with ∪, ∩, complement forms a Boolean algebra.Source: Logic
🎲Sample space in probability: events often form a σ-algebra, related to power set.Source: Probability
📊P(∅) = {∅}. The empty set has exactly one subset: itself.Source: Edge case
🔗Cantor used |P(ℕ)| > |ℕ| to prove uncountability of the reals.Source: Set theory

📖 How It Works

P(S) contains every possible subset of S. For n elements, 2^n subsets because each element is independently in or out. Enumerate by i = 0..2^n−1: subset i includes element j iff bit j of i is 1.

Proper subsets exclude S itself: 2^n − 1. k-element subsets: C(n,k).

📝 Worked Example: P({a,b,c})

Step 1: |S| = 3, so |P(S)| = 2³ = 8

Step 2: Binary 000→∅, 001→{c}, 010→{b}, 011→{b,c}, 100→{a}, 101→{a,c}, 110→{a,b}, 111→{a,b,c}

Result: P(S) = {∅, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}}

🚀 Real-World Applications

💻 Subset Sum

NP-complete: find subset summing to target.

📊 Feature Selection

2^n subsets of n features to evaluate.

🔐 Access Control

Power set of permissions for roles.

📐 Boolean Algebra

Power set with union, intersection.

🎲 Probability

Event space as σ-algebra on power set.

🔬 Logic

Truth tables, propositional logic.

⚠️ Common Mistakes to Avoid

  • Forgetting ∅: The empty set is always in P(S).
  • Confusing proper vs full: Proper excludes S; full includes it.
  • n > 10: 2^n grows fast. Limit n for enumeration.
  • Duplicate elements: Sets have no duplicates; we deduplicate input.
  • Order in sets: {a,b} = {b,a} — order doesn't matter.

🎯 Expert Tips

💡 Binary Trick

Loop i=0 to 2^n-1; for each j, if (i>>j)&1, include element j.

💡 Large Sets

For n>15, 2^n is huge. Use combinatorial counting instead of enumeration.

💡 Proper vs Full

Proper subsets exclude S. Count = 2^n − 1.

💡 C(n,k) Sum

Σ C(n,k) = 2^n. Each subset has some size k.

📊 Reference Table

n|P(S)|C(n,0)..C(n,n)
011
121, 1
241, 2, 1
381, 3, 3, 1
4161, 4, 6, 4, 1
5321, 5, 10, 10, 5, 1

📐 Quick Reference

2^n
|P(S)|
2^n − 1
Proper subsets
C(n,k)
k-element subsets
P(∅)={∅}
Empty set

🎓 Practice Problems

P({1,2}) → 4 subsets
|P({a,b,c,d})| → 16
Proper subsets of 3-element set → 7
2-element subsets of 5-element set → C(5,2)=10

❓ FAQ

What is the power set?

P(S) is the set of all subsets of S. Includes ∅ and S. |P(S)| = 2^|S|.

Why 2^n subsets?

Each of n elements is either in or out. 2 choices per element ⇒ 2^n total.

What are proper subsets?

Subsets of S not equal to S. Count = 2^n − 1.

How does binary encoding work?

Number 0 to 2^n−1: bit j = 1 means element j is in the subset.

What is Cantor's theorem?

For any set S, |P(S)| > |S|. No set equals its power set.

Applications in CS?

Subset sum, knapsack, feature selection, exhaustive search.

P(∅)?

P(∅) = {∅}. The empty set has one subset: itself.

📌 Summary

P(S) = set of all subsets. |P(S)| = 2^n. Binary encoding: integer i ↔ subset with bit j = 1 iff element j included. Proper subsets: 2^n − 1. k-element subsets: C(n,k).

✅ Verification Tip

Verify: Σ C(n,k) for k=0..n = 2^n. Check |P(∅)| = 1, |P({x})| = 2.

🔗 Next Steps

Explore the Combination Calculator for C(n,k), Subset Calculator, or Binomial Coefficient Calculator.

⚠️ Disclaimer: For n > 10, 2^n > 1024. Display is limited for performance.

HI THERE
👋Power sets model all possible combinations — used in logic, topology, and computer science.
AI
NumberVibe

Related Calculators

Subset Calculator

Check if set A is subset of B (A⊆B), proper subset (A⊂B), or superset. List all subsets (power set), count subsets. Find intersection, union, complement. Bar...

Mathematics

Binomial Coefficient Calculator

Calculate binomial coefficients C(n,k) = n!/(k!(n-k)!) — the number of ways to choose k items from n without regard to order. Also known as n choose k or ⁿCₖ. Features Pascal's triangle row generation, factorial breakdown, Bar chart of full row C(n,0)..C(n,n), Doughnut chart showing k vs n−k. Applications: lottery (C(49,6)), poker hands (C(52,5)), committee selection, DNA combinations, binary strings. Educational content on combinatorics, binomial theorem (x+y)ⁿ expansion, and Pascal's identity.

Mathematics

Absolute Value Equation Calculator

Solve absolute value equations and inequalities including |ax+b|=c, |ax+b|=|cx+d|, and |ax+b|≤c forms. Features solution verification, number line...

Mathematics

Absolute Value Inequalities Calculator

Solve absolute value inequalities |ax+b| < c, > c, ≤ c, ≥ c. Get solution intervals, interval notation, set builder notation, number line visualization...

Mathematics

Bessel Function Calculator

Calculate Bessel functions J_n(x), Y_n(x), I_n(x), and K_n(x) with series approximation. Supports first kind (J), second kind/Neumann (Y), and modified Bessel functions (I, K). Applications include vibrating drum modes, electromagnetic waveguides, heat conduction in cylinders, and Bessel filter design in signal processing. Uses the power series J_n(x) = Σ (-1)^m/(m!(m+n)!)·(x/2)^(2m+n). Includes convergence info, Bar chart comparing values at different x, Doughnut chart showing series term contributions, and educational content on Bessel's differential equation and cylindrical harmonics.

Mathematics

Box Method Calculator

Multiply polynomials visually using the box method (area model). Step-by-step solutions for binomials, trinomials, and factoring. Interactive grid...

Mathematics