Power Set Calculator
Result
Power Set Properties
Definition
The power set of a set S is the set of all possible subsets of S, including the empty set and S itself.
P(S) = {A | A ⊆ S}
Properties
- Cardinality: |P(S)| = 2^n, where n is the number of elements in S
- Always includes the empty set ∅
- Always includes the original set S
- Each element can be either included or excluded in a subset
Example Problems
Example 1: Set {a, b}
Power Set:
- ∅ (empty set)
- {a}
- {b}
- {a, b}
Total subsets: 2² = 4
Example 2: Set {1, 2, 3}
Power Set:
- ∅
- {1}
- {2}
- {3}
- {1, 2}
- {1, 3}
- {2, 3}
- {1, 2, 3}
Total subsets: 2³ = 8
Binary Representation
Each subset can be represented by a binary number where:
- 1 means the element is included
- 0 means the element is excluded
Example for set {a, b, c}:
- 000 = ∅
- 001 = {c}
- 010 = {b}
- 011 = {b, c}
- 100 = {a}
- 101 = {a, c}
- 110 = {a, b}
- 111 = {a, b, c}
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