1. What is a Power Set?

The power set of any set S is the set of all possible subsets of S, including the empty set and S itself. For a set with n elements, its power set contains 2^n elements.

2. How Does the Calculator Work?

The calculator generates all possible combinations of the input elements to form the power set using the formula:

Where:

• \( P(S) \) — Power set of S
• \( X \) — Any subset of S
• \( \subseteq \) — Subset relation

Explanation: The algorithm builds the power set incrementally by considering each element and adding it to all existing subsets.

3. Mathematical Properties

Details:

• Cardinality: For a set with n elements, the power set has 2^n elements
• The power set forms a Boolean algebra under the operations of union, intersection, and complement
• Power sets are fundamental in topology, measure theory, and other areas of mathematics

4. Using the Calculator

Tips: Enter elements separated by commas. Duplicates will be removed. The empty set is always included in the power set.

5. Frequently Asked Questions (FAQ)

Q1: What's the power set of an empty set?
A: The power set of the empty set is {∅} - a set containing only the empty set.

Q2: How many subsets does a set with 5 elements have?
A: 2^5 = 32 subsets, including the empty set and the set itself.

Q3: What's the difference between a subset and a proper subset?
A: A proper subset is any subset that is not equal to the original set.

Q4: Can this calculator handle large sets?
A: For practical purposes, sets with more than 15-20 elements may be too large to display effectively.

Q5: How is the power set used in computer science?
A: Power sets are used in algorithms, state machines, and representing all possible combinations.