Power Set Calculator

Generate all subsets of a set and explore power set properties with step-by-step solutions.

Enter Set Elements

Enter up to 10 elements separated by commas. Duplicates will be removed.

Result

Number of Subsets
8
2^3 = 8
Original Set {a, b, c}
Set Cardinality |S| 3
Power Set Cardinality |P(S)| 8
Proper Subsets 7

Step-by-Step Solution

P(S) = 2^|S| = 2^3 = 8 subsets

All Subsets:

Understanding Power Sets

In mathematics, the power set of a set S, denoted P(S) or 2S, is the set of all subsets of S, including the empty set and S itself. If S has n elements, then P(S) has 2n elements. The power set is a fundamental concept in set theory, combinatorics, and computer science.

How to Generate a Power Set

The power set can be generated using several methods. The binary counting method maps each subset to a binary number from 0 to 2n-1, where each bit position determines whether the corresponding element is included. Alternatively, the recursive method builds subsets by choosing whether to include or exclude each element.

Power Set Size

The number of subsets is always a power of 2.

|P(S)| = 2^|S|

Empty Set

The empty set is always a member of every power set.

{} is in P(S)

Proper Subsets

The number of proper subsets excludes the set itself.

Proper subsets = 2^n - 1

k-Element Subsets

The number of subsets with exactly k elements.

C(n, k) = n! / (k!(n-k)!)

Applications of Power Sets

  • Boolean algebra: The power set forms a Boolean algebra under union, intersection, and complement.
  • Combinatorics: Counting problems often involve enumerating subsets of a given set.
  • Computer science: Power sets model possible states in search algorithms, database queries, and feature selection.
  • Topology: Power sets are used to define topologies on sets.
  • Probability: The power set of the sample space defines all possible events.

Relationship to Binary Numbers

There is a natural one-to-one correspondence between subsets of an n-element set and n-bit binary numbers. For a set {a, b, c}, the binary number 101 corresponds to the subset {a, c} (first and third elements included). This bijection is why the power set has exactly 2n elements and provides an efficient algorithm for enumeration.

Cantor's Theorem

Georg Cantor proved that for any set S, the power set P(S) has strictly greater cardinality than S itself. This holds even for infinite sets, which means there is no largest cardinal number. This result is one of the foundational theorems of set theory and has profound implications for the philosophy of mathematics.