Updated Advanced Set Tool

Set Theory Calculator

Compute union, intersection, difference, symmetric difference, complement, power set, Cartesian product, relations and cardinality for sets of numbers or symbols. Enter sets as comma-separated lists and explore core operations from set theory and discrete mathematics.

Union & Intersection Difference & Complement Power Set & Cartesian Product Relations & Cardinality

Work with Finite Sets Using Comma-Separated Inputs

Enter two sets A and B as comma-separated lists. Optionally, enter a universal set U to compute complements. The same base sets are used across all tabs, so you can explore basic operations, power sets, Cartesian products, relations and cardinality without retyping the data.

Use commas to separate elements, for example: 1, 2, 3 or a, b, c. Duplicates are removed automatically and sets are normalized internally. Elements are treated as strings, so 1 and "1" are considered the same element.

The power set ℘(A) is the set of all subsets of A. The number of subsets grows as 2n, so this calculator limits the size of A for performance and readability.

The Cartesian product A × B is the set of all ordered pairs (a, b) with a ∈ A and b ∈ B. For large sets, the number of pairs can be big, so the calculator may truncate the list for display.

Consider a relation R on the set A. Enter R as a list of ordered pairs using parentheses, for example: (1,1), (1,2), (2,2). The calculator checks whether R is reflexive, symmetric, antisymmetric and transitive on A, and whether it defines an equivalence relation.

This tab summarizes cardinalities and basic relationships between A and B: subset, proper subset, equality and disjointness. It also computes |A ∪ B|, |A ∩ B|, |A − B| and |B − A|.

Set Theory Calculator – Complete Guide to Set Operations, Power Set, Cartesian Product and Relations

The Set Theory Calculator on MyTimeCalculator is built for students, teachers, computer science learners and anyone who works with finite sets. It brings together the most important operations from set theory in a single interface: union, intersection, difference, symmetric difference, complement, power set, Cartesian product, relations and cardinality checks.

Instead of doing each operation by hand, you can enter sets A and B once and explore how they interact under different operations. The calculator removes duplicates automatically and treats elements consistently, allowing you to focus on the concepts rather than bookkeeping.

1. Sets, Elements and Notation

A set is a collection of distinct objects, called elements. In notation, we often write A = {1, 2, 3} or B = {a, b, c}. Sets can contain numbers, symbols or any other distinct labels. This calculator works with finite sets entered as comma-separated lists such as:

A = 1, 2, 3
B = 3, 4, 5

Internally, the tool converts these lists into sets by trimming extra spaces and removing duplicate values. This matches the mathematical idea that ordering and repeated entries do not matter in a set.

2. Basic Set Operations: Union, Intersection, Difference and Symmetric Difference

The basic operations in the first tab are the building blocks of set theory:

  • Union (A ∪ B): the set of elements that belong to A or B or both.
  • Intersection (A ∩ B): the set of elements that belong to both A and B.
  • Difference (A − B): elements in A that are not in B.
  • Difference (B − A): elements in B that are not in A.
  • Symmetric Difference (A △ B): elements that belong to exactly one of A or B, but not both.

These operations appear in many contexts: database queries using logical conditions, probability of combined events, classification tasks and logic circuits. The calculator computes all of them at once, so you can quickly compare the results for the same underlying sets.

3. Complements with Respect to a Universal Set

In many problems, we work relative to a universal set U that contains all elements under consideration. The complement of A, written Ac, is the set of elements in U that are not in A:

Ac = U − A.

In this calculator, if you provide a universal set U, the Basic Operations tab computes complements such as U − A and U − B. This is especially helpful in probability, logic and Venn diagram problems where you want to understand what lies outside a given event or condition.

4. Power Set ℘(A)

The power set of A, written ℘(A), is the set of all subsets of A. If A has n elements, then ℘(A) has 2n elements. For example, if A = {1, 2}, then:

℘(A) = { ∅, {1}, {2}, {1, 2} }.

Because the number of subsets grows exponentially, the calculator applies a safety limit to the size of A before listing every subset. When A is small, you can see all subsets written in a clear format. When A is larger, the tool focuses on |A| and |℘(A)| so that you still see the combinatorial side of the problem without generating an overwhelming list.

5. Cartesian Product A × B

The Cartesian product A × B is the set of all ordered pairs (a, b) with a ∈ A and b ∈ B. It is central in coordinate geometry, relations, functions and discrete mathematics:

A × B = { (a, b) | a ∈ A, b ∈ B }.

If |A| = m and |B| = n, then |A × B| = m·n. The Cartesian Product tab reports |A|, |B| and |A × B|, and it lists the pairs up to a reasonable limit. This lets you see the structure of product sets without scrolling through hundreds of entries.

6. Relations on A and Their Properties

A relation R on a set A is a subset of A × A. Each element of R is an ordered pair (a, b). Properties of relations are important in discrete mathematics and computer science:

  • Reflexive: for every a in A, (a, a) is in R.
  • Symmetric: whenever (a, b) is in R, (b, a) is also in R.
  • Antisymmetric: if (a, b) and (b, a) are in R, then a = b.
  • Transitive: whenever (a, b) and (b, c) are in R, (a, c) is also in R.

A relation that is reflexive, symmetric and transitive on A is called an equivalence relation. It partitions A into disjoint equivalence classes. In the Relations tab, you enter R as a list of pairs such as (1,1), (1,2), (2,2), and the calculator tests all four properties and reports whether R is an equivalence relation.

7. Cardinality and Set Relationships

Cardinality is the number of elements in a set and is denoted |A|. The Cardinality & Checks tab shows:

  • |A| and |B|
  • |A ∪ B| and |A ∩ B|
  • |A − B| and |B − A|
  • Whether A is a subset of B (A ⊆ B)
  • Whether A is a proper subset of B (A ⊂ B)
  • Whether A and B are equal
  • Whether A and B are disjoint (empty intersection)

These checks are useful both in pure set theory and in applied settings such as classifying data, reasoning about overlapping categories and evaluating conditions in algorithms.

8. How to Use the Set Theory Calculator Effectively

  1. Enter Set A and Set B as comma-separated lists at the top of the calculator.
  2. Optionally enter a universal set U for complement calculations.
  3. Use the Basic Operations tab to see A ∪ B, A ∩ B, A − B, B − A, A △ B and complements.
  4. Use the Power Set tab to explore ℘(A) for small sets.
  5. Use the Cartesian Product tab to generate A × B and see its size.
  6. Use the Relations tab to define a relation R on A and analyze its properties.
  7. Use the Cardinality tab to summarize set sizes and relationships such as subset and disjointness.

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Set Theory Calculator FAQs

Frequently Asked Questions

Answers to common questions about entering sets, computing operations and interpreting results in this Set Theory Calculator.

Enter sets as comma-separated lists, such as 1, 2, 3 or a, b, c. Spaces are ignored, and duplicate values are removed automatically. The order of elements does not matter. The same Set A and Set B are reused across all tabs so you can explore multiple operations quickly.

Yes. The calculator treats elements as string labels, so you can use numbers, letters or other short tokens. For example, 1, 2, a, b is allowed. Just make sure your elements are separated by commas and that you do not rely on hidden spaces to distinguish them, since trimming is applied.

In set theory, a set cannot contain the same element twice. Writing A = {1, 1, 2} is equivalent to A = {1, 2}. To match this definition, the calculator automatically removes duplicates. This ensures that cardinalities, unions and intersections reflect true set behavior rather than list behavior.

The number of subsets grows as 2n, so even moderate n can produce a very long list. To keep the output readable and responsive, the calculator enforces a maximum size for A when listing every subset. If A is too large, it will still compute |A| and |℘(A)| but may not display all subsets explicitly.

The calculator can only compute complements when a universal set U is provided. If U is left empty, the Basic Operations tab still computes union, intersection, differences and symmetric difference, but complement values will not be reported. In many problems the choice of U depends on context, so you have full control over how it is defined.

First enter Set A at the top of the calculator. Then in the Relations tab, write R as a comma-separated list of ordered pairs using parentheses, for example: (1,1), (1,2), (2,2). The calculator parses these pairs, checks that they involve elements of A and evaluates whether R is reflexive, symmetric, antisymmetric and transitive on A. It also reports whether R is an equivalence relation.

The number of pairs in A × B is |A|·|B|. When this number is large, the full list can become long and difficult to read. To preserve performance and readability, the calculator may truncate the displayed list while still reporting the full cardinality |A × B|. This is common when both sets have many elements.

Yes. The tool is ideal for verifying hand calculations of union, intersection, differences, complements, power sets, Cartesian products and relation properties. It can also help build intuition by letting you experiment with many different examples quickly. For graded assessments, always follow your instructor’s rules on the use of calculators and online tools.