Subset Calculator
Generate all subsets of a set and calculate subset counts with proper mathematical notation
Calculate Set Subsets
Enter distinct numbers only. Duplicates will be automatically removed.
Subset Calculation Results
Mathematical Formulas
Example Calculation
Example: Set A = 3
Set elements: n = 3
Total subsets: 2³ = 8
Proper subsets: 2³ - 1 = 7
All subsets: ∅, 1, 2, 3, 2, 3, 3, 3
Subset Count by Size
Size 0: C(3,0) = 1 subset → ∅
Size 1: C(3,1) = 3 subsets → 1, 2, 3
Size 2: C(3,2) = 3 subsets → 2, 3, 3
Size 3: C(3,3) = 1 subset → 3
Total: 1 + 3 + 3 + 1 = 8 subsets
Types of Subsets
Subset
A ⊆ B: All elements of A are in B
Includes the set itself
Proper Subset
A ⊊ B: A ⊆ B and A ≠ B
Excludes the set itself
Empty Set
∅: Subset of every set
Contains no elements
Quick Facts
Every set has at least one subset: ∅
The empty set has no proper subsets
A set with n elements has 2ⁿ subsets
Pascal's triangle shows subset counts by size
Understanding Subsets and Power Sets
What is a Subset?
A subset is a set where every element also belongs to another set. If A is a subset of B, then all elements of A are contained in B, but B may have additional elements.
Subset vs Proper Subset
- •Subset (⊆): Includes the set itself as a subset
- •Proper subset (⊊): Excludes the set itself
- •Empty set (∅): Proper subset of all non-empty sets
Mathematical Formulas
Total subsets = 2ⁿ
Proper subsets = 2ⁿ - 1
k-element subsets = C(n,k)
n: Number of elements in the original set
k: Number of elements in the subset
C(n,k): Binomial coefficient "n choose k"
Power Set: The set of all subsets of a given set, denoted as P(A).
Applications
Probability
Counting favorable outcomes in sample spaces
Combinatorics
Selecting items from larger collections
Computer Science
Algorithm design and data structures