Power Set Calculator
Generate the power set (set of all subsets) for any given set with step-by-step explanation
Set Input
Enter up to 10 unique elements separated by commas
Original Set:
Power Set Results
Power Set P(S) - Grouped by Subset Size:
Subsets of size 0:
Subsets of size 1:
Subsets of size 2:
Subsets of size 3:
Complete Power Set P(S):
Step-by-Step Generation Process
Binary Method Explanation:
Each subset corresponds to a binary number from 0 to 2^n - 1, where each bit position indicates whether to include that element.
Example Sets
Small Set
Set: {A, B}
Power Set: {∅, {A}, {B}, {A,B}}
Cardinality: 2^2 = 4
Medium Set
Set: {1, 2, 3, 4}
Cardinality: 2^4 = 16 subsets
Power Set Properties
Always contains the empty set ∅
Always contains the original set itself
Cardinality is always 2^n where n is the original set size
Number of k-element subsets is C(n,k)
Power Set Tips
Power set grows exponentially (2^n)
Empty set is a subset of every set
Use binary representation to systematically generate subsets
Power set of power set has 2^(2^n) elements
Understanding Power Sets
What is a Power Set?
The power set of a set S, denoted P(S) or 2^S, is the set of all subsets of S, including the empty set ∅ and S itself. It's called a "power set" because if S has n elements, then P(S) has 2^n elements.
Mathematical Definition
P(S) = {A : A ⊆ S}
The set of all sets A such that A is a subset of S
Key Properties
- •|P(S)| = 2^|S| (cardinality formula)
- •∅ ∈ P(S) for any set S
- •S ∈ P(S) for any set S
- •If A ⊆ B, then P(A) ⊆ P(B)
Generation Algorithm
- 1
Binary Representation: Use numbers 0 to 2^n - 1
- 2
Convert to Binary: Each number becomes a binary string
- 3
Map to Elements: 1 means include element, 0 means exclude
- 4
Generate Subset: Create subset based on binary pattern
Example Process
Set: {A, B, C}
000 → ∅
001 → {C}
010 → {B}
011 → {B, C}
100 → {A}
101 → {A, C}
110 → {A, B}
111 → {A, B, C}
Applications
- • Computer science algorithms
- • Boolean algebra and logic
- • Database query optimization
- • Combinatorics and probability
- • Graph theory and networks