Power Set Calculator
Enter up to ten distinct real numbers to generate every subset, count the power set, and inspect how subsets are distributed by size.
Set elements
Distinct numbers only; use a period for decimals.
Required in a non-empty calculation.
Optional; leave blank when unused.
Optional; every value must be unique.
Optional; negative values are allowed.
Optional; examples include 5 or 2.5.
Optional; exponential notation is rejected.
Optional; do not use grouping commas.
Optional; zero is a valid element.
Optional; values may be fractional.
Optional; ten elements produce 1,024 subsets.
Result
Updates automatically as valid inputs change.
Power set cardinality
2⁴ = 16
Original set
Proper subsets
Subsets by size
Counts follow the binomial coefficients C(n, k).
| Subset size (k) | Number of subsets | Share of power set |
|---|
The row counts sum to the total power set cardinality. Complementary sizes have equal counts: C(n, k) = C(n, n – k).
Complete power set
Every subset is grouped by its number of elements.
| Subset size | Count | Subsets |
|---|
The empty subset is written as ∅. The final row always includes the original set itself.
How to use the power set calculator
What this calculator does
This calculator constructs the power set of a finite set containing up to ten distinct real numbers. A power set is the collection of every possible subset, including the empty subset and the original set. The calculator reports the number of elements, the exact power set cardinality, the count of proper subsets under the convention used here, a size-by-size distribution, and the full subset list. It is an exact combinatorial calculation for the values entered; it does not infer meaning, probability, order, or frequency from those values. For a formal definition and examples with nested braces, see the LibreTexts lesson on subsets and power sets.
When to use it
Use the calculator when checking a discrete-mathematics exercise, verifying that a hand-written power set is complete, enumerating every possible selection from a small group, or studying how subset counts are distributed by size. It is especially useful for examples with four to ten elements, where manual listing becomes error-prone even though the counting formula remains simple.
How to calculate
- The calculator opens with the demonstration set {1, 2, 3, 4}. Its 16-subset result and a validated Excel workbook are ready immediately.
- Replace the values in Element 1 through Element 10. Enter one number per field and leave unused optional fields blank. Results update live; no Calculate button is required.
- Read Power set cardinality first, then use Subsets by size to verify the binomial counts and Complete power set to inspect every subset.
- Select Download Excel to export the current inputs, results, distribution, and one-row-per-subset listing as a real .xlsx workbook.
- Select Reset to clear the demonstration values, results, validation state, and workbook cache. Download Excel is then disabled until at least one complete valid value is entered again.
Input guide
Element 1 is required for a non-empty calculation after Reset. Element 2 through Element 10 are optional. Each field accepts a finite real number written with digits, an optional leading plus or minus sign, and an optional period as the decimal separator; examples include 7, -3, 0, and 2.5. Grouping commas, decimal commas, units, fractions such as 1/2, and scientific notation such as 1e3 are deliberately rejected so that pasted input cannot be silently reinterpreted. Every entered number must be distinct: entering 2 and 2.0 represents the same set element and produces a duplicate-value error. Adding one valid distinct element doubles the total number of subsets because that new element can be either excluded from or included in each existing subset. The tenth field is the practical limit because ten elements already produce 1,024 subsets.
Output guide
Set size equals the number of entered elements. All subsets and the main Power set cardinality result equal 2 raised to that number. Formula shows the exact exponent and result. Original set repeats the normalized numeric inputs in their entered order. Proper subsets excludes the two improper subsets – the empty subset and the original set – so it is 2ⁿ – 2 for a non-empty set. The Subsets by size table lists subset size k, the exact count C(n, k), and that count as a percentage of the whole power set. The Complete power set table groups all subsets by k and uses ∅ for the empty subset. These are exact identities, not statistical estimates. The MathWorld finite-set reference connects the C(n, k) row counts to the identity whose sum is 2ⁿ.
Worked example
For the startup values 1, 2, 3, and 4, the set size is n = 4. Each of the four elements has two independent states in a subset: included or excluded. Therefore the calculator evaluates 2 × 2 × 2 × 2 = 2⁴ = 16. The size distribution is 1 subset of size 0, 4 of size 1, 6 of size 2, 4 of size 3, and 1 of size 4. Those counts sum to 16. Under this calculator's improper-subset convention, ∅ and {1, 2, 3, 4} are excluded from the proper-subset count, leaving 14 proper subsets. The first row of the complete list is ∅, while the final row contains the original four-element set.
Why the answer grows so quickly
The exponential growth comes from independent yes-or-no choices. With n elements, a subset-making procedure makes n binary decisions, producing 2ⁿ distinct inclusion patterns. Five elements produce 32 subsets, eight produce 256, and ten produce 1,024. This is why the calculator limits the input to ten numbers: the formula is easy to evaluate for much larger n, but explicitly rendering every subset becomes increasingly bulky. The MathWorld power-set overview summarizes the definition, notation, and 2ⁿ cardinality rule.
Common mistakes
- Forgetting the empty subset or the original set. Both belong to every power set.
- Treating repeated values as separate elements. A set contains distinct elements, so duplicates do not create new subsets.
- Counting only non-empty subsets. The total is 2ⁿ; the non-empty total is 2ⁿ – 1.
- Confusing subset size with the number of subsets. C(n, k) counts how many subsets contain exactly k elements.
- Assuming input order changes the mathematical set. It only changes the display order used inside each generated subset.