What is the union of two sets?

The union of sets A and B (written A U B) is the set of all elements that are in A, in B, or in both. For example, {1, 2, 3} U {3, 4, 5} = {1, 2, 3, 4, 5}. Union combines both sets without duplicates.

What is the intersection of two sets?

The intersection of sets A and B (written A ∩ B) is the set of elements that are in both A and B. For example, {1, 2, 3} ∩ {3, 4, 5} = {3}. If two sets have no common elements, their intersection is the empty set.

The power set P(A) of a set A is the set of all possible subsets of A, including the empty set and A itself. If A has n elements, the power set has 2^n subsets. For example, P({1, 2}) = {{}, {1}, {2}, {1, 2}}.

Set Operations Calculator - Union, Intersection, Power

Q: What is a set in mathematics?

A set is a well-defined collection of distinct objects called elements. Sets are denoted with curly braces, e.g., A = {1, 2, 3}. The order of elements does not matter, and duplicates are ignored. Sets are fundamental to discrete mathematics, logic, and computer science.

Q: What is the symmetric difference?

The symmetric difference of A and B (written A Δ B) is the set of elements in either A or B, but not in both. It equals (A - B) U (B - A). For example, {1, 2, 3} Δ {3, 4, 5} = {1, 2, 4, 5}.

Set Operations Calculator

Compute union, intersection, difference, symmetric difference, complement, power set, and cardinality. Enter sets as comma-separated values and see all operations at once.

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Set Operations

Set A (comma-separated)

Set B (comma-separated)

Universal Set (optional, for complement)

Results

A ∪ B (Union) —

A ∩ B (Intersection) —

A − B (Difference) —

B − A (Difference) —

A Δ B (Symmetric Diff) —

A' (Complement) —

B' (Complement) —

Cardinality — —

P(A) (Power Set) —

A ⊆ B? —

B ⊆ A? —

Disjoint? —

A = B? —

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How to Use the Set Operations Calculator

Enter the elements of Set A and Set B as comma-separated values. Optionally, define a Universal Set for complement operations. All operations are computed simultaneously and displayed in a clear grid. The calculator handles both numeric and text elements.

For example, with A = {1, 2, 3, 4} and B = {3, 4, 5, 6}: Union = {1, 2, 3, 4, 5, 6}, Intersection = {3, 4}, A − B = {1, 2}, Symmetric Difference = {1, 2, 5, 6}.

Understanding Set Operations

Sets are one of the most fundamental concepts in mathematics. A set is simply a collection of distinct objects. Set operations provide ways to combine, compare, and analyze these collections.

Core Operations

Union (A ∪ B) combines all elements from both sets. Intersection (A ∩ B) finds common elements. Difference (A − B) finds elements in A but not B. Symmetric difference (A Δ B) finds elements in exactly one set. Complement (A') finds elements in the universal set but not in A.

Power Set

The power set P(A) is the set of all subsets of A. If A has n elements, P(A) has 2ⁿ subsets. The power set grows exponentially, which is why this calculator limits it to sets with up to 10 elements for display.

Applications

Set operations are used in database queries (SQL UNION, INTERSECT, EXCEPT), probability theory (event spaces), data analysis (Venn diagram analysis), programming (array operations, data filtering), formal logic, and combinatorics. Understanding sets is fundamental to discrete mathematics and computer science.

Frequently Asked Questions

What is a set in mathematics?

A collection of distinct objects called elements. Order does not matter and duplicates are ignored.

What is the union?

A ∪ B contains all elements from either A or B or both.

What is the intersection?

A ∩ B contains only elements present in both A and B.

What is the power set?

The set of all subsets of A, including the empty set and A itself. Has 2ⁿ elements.

What is the symmetric difference?

A Δ B contains elements in either A or B but not both. Equals (A − B) ∪ (B − A).

Disclaimer: This calculator is for informational and educational purposes only. Results are estimates and should not be considered professional expert advice. Consult a qualified professional before making decisions based on these calculations. See our full Disclaimer.