What is the union of two sets?

The union, A ∪ B, is the set of every element that appears in A, in B, or in both — with duplicates removed. It's the combined collection of everything in either set.

What is the intersection of two sets?

The intersection, A ∩ B, is the set of elements that appear in both A and B. If the sets share nothing in common, the intersection is the empty set.

What's the difference between A − B and symmetric difference?

A − B contains only elements in A that are not in B (one direction). Symmetric difference (A △ B) contains elements in either set but not in both — everything except the overlap.

What is a Cartesian product?

The Cartesian product A × B is the set of every possible ordered pair (a, b) where a comes from A and b comes from B. If A has m elements and B has n elements, A × B has m × n pairs.

The power set P(A) is the set of all possible subsets of A, including the empty set and A itself. A set with n elements has exactly 2ⁿ subsets, so power sets grow very quickly with set size.

What is a complement, and why do I need a universal set?

The complement of A, written Aᶜ, is everything in a defined universal set U that is NOT in A. Since 'everything not in A' is meaningless without a boundary, you must specify U to compute a complement.

Are duplicate elements removed automatically?

Yes. Sets by definition contain no duplicates, so entering the same element twice in a field only counts it once.

Why is there a size limit on the power set operation?

A power set has 2ⁿ subsets, so a 20-element set would already produce over a million subsets. The calculator caps power sets at 10 elements (1,024 subsets) to keep the result readable and performant.

Set Theory Calculator — Union, Intersection, Difference, and Power Set

Our Set Theory Calculator performs every core set operation on custom sets you enter: union (A ∪ B), intersection (A ∩ B), difference (A − B), symmetric difference (A △ B), Cartesian product (A × B), power set (P(A)), and complement (Aᶜ relative to a universal set). Enter your elements separated by commas and get an instant, correctly formatted result with an explanation of how it was derived.

Quick Answer

This calculator performs every core set operation — union, intersection, difference, symmetric difference, Cartesian product, power set, and complement — on sets you enter as comma-separated elements, returning a correctly formatted result with an explanation of how it was derived.

Set A (comma-separated)

Set B (comma-separated)

Enter your set(s), then click Calculate.

How to Use the Set Theory Calculator — Union, Intersection, Difference & More

1
Choose a set operation from the button row.
2
Enter the elements of Set A, separated by commas.
3
For operations that need a second set or a universal set, fill in Set B or the Universal Set field.
4
Click 'Calculate' to see the resulting set and a short explanation of how it was derived.

Why Use Set Theory Calculator — Union, Intersection, Difference & More?

Set operations are the building blocks of discrete mathematics, database query logic (SQL joins mirror set operations directly), and probability theory. Working them out by hand for anything beyond a handful of elements is tedious and error-prone — it's easy to accidentally include or drop an element, especially with symmetric difference or power sets, where the number of subsets doubles with every added element. This calculator computes the exact result for any operation instantly and shows the reasoning, so you can check your work or explore how each operation behaves.

Frequently Asked Questions

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