This power set calculator, provided by Hesapstan, helps you enter a finite set, generate all of its subsets, see the 2^n count, and inspect the subset list grouped by size when the set is small enough to display clearly.
What does this power set calculator calculate?
The calculator reads a comma-separated finite set, removes repeated elements as set duplicates, and generates the power set: the set of all possible subsets.
- The cleaned version of your input set
- The cardinality n of the set
- The total number of subsets, 2^n
- The number of proper subsets, 2^n − 1
- A grouped subset listing when n≤10
If your main goal is to see every subset, this power set page is the direct tool. If you only need subset counts or C(n,k), the subset calculator is often the cleaner fit.
What is a power set?
The power set of a set A is the set whose elements are all subsets of A. The empty set is included, and the original set A itself is also included.
For A={a,b,c}, each element has two choices: included or not included. That gives 2×2×2=8 subsets.
- Choose no elements: ∅
- Choose one element: {a}, {b}, {c}
- Choose two elements: {a,b}, {a,c}, {b,c}
- Choose all three elements: {a,b,c}
A subset is a possible selection from the original set. Selecting nothing is still a valid selection, so ∅ belongs to every power set.
Why is the number of subsets 2^n?
A set with n elements has 2^n subsets because each element creates one independent yes/no decision: include it in the subset or leave it out.
The number of proper subsets is 2^n − 1 because the original set is counted as a subset, but it is not counted as a proper subset.
- n=0 gives one subset: ∅.
- n=3 gives 8 subsets.
- n=10 gives 1024 subsets, which is why the calculator caps visible listing at n≤10.
At n=11 there are 2048 subsets, and at n=12 there are 4096. The n≤10 listing cap is a readability and page-usability limit, not a mathematical limitation.
How to use the calculator
- Enter set elements separated by commas, such as a, b, c.
- Check the cleaned set if duplicates were removed.
- Read n, 2^n total subsets and 2^n − 1 proper subsets.
- If n≤10, open the grouped subset lists by size.
In set theory, {a,a,b} is the same set as {a,b}. The calculator deduplicates repeated entries and shows a visible note rather than silently inflating the result.
Element labels may be numbers, letters or short text values. What matters is that each comma-separated item represents one distinct element in your intended set.
Worked example: power set of {a,b,c}
For A={a,b,c}, the set has n=3 elements. The total number of subsets is 2^3=8, and the number of proper subsets is 2^3−1=7.
- Size 0: ∅
- Size 1: {a}, {b}, {c}
- Size 2: {a,b}, {a,c}, {b,c}
- Size 3: {a,b,c}
The power set is not the same thing as the original set. It is a new set whose members are the eight subsets listed above.
The original set has 3 elements. Its power set has 8 elements, and each of those 8 elements is itself a subset.
Power set, subset and proper subset
A subset is one possible selection from a set. The power set is the collection of all such selections. A proper subset is any subset except the original set itself.
- Subset: one choice, such as {a,b}.
- Power set: the set containing all choices, such as ∅, {a}, {b}, {a,b}.
- Proper subset: every subset except the original full set.
- k-subset: a subset with exactly k elements, usually counted with C(n,k).
Use the power set calculator to display all subsets. Use the subset calculator when you primarily need counts such as total subsets, proper subsets or k-subsets.
Common mistakes and limits
- Forgetting the empty set.
- Forgetting that the original set is also a subset.
- Counting repeated input entries as new elements.
- Using n^2 instead of 2^n.
- Expecting a readable full listing when n is larger than 10.
This tool works with finite comma-separated element lists. Infinite sets, interval notation, and set-builder conditions are outside this calculator's scope.
Frequently Asked Questions
What is a power set?
A power set is the set of all subsets of a given set.
How many subsets does a set with n elements have?
It has 2^n subsets.
Is the empty set included in the power set?
Yes. The empty set is a subset of every set, so it appears in every power set.
Is a power set the same as a subset?
No. A subset is one selection; the power set is the collection of all possible selections.
Why does the calculator stop listing after n=10?
Because the number of subsets grows exponentially. n=10 already produces 1024 subsets, which is a practical display limit.