# Set operations calculator

> Union, intersection, differences and symmetric difference of two sets of numbers or words, with the Jaccard index and a Venn diagram.

Interactive version: https://www.calcopenly.com/math/set-operations-calculator
Subject: Math calculators

The calculator treats each list as a set, dropping repeats, and compares the two. The union A ∪ B holds everything in either set, the intersection A ∩ B what is in both, the differences A − B and B − A what is in one but not the other, and the symmetric difference A △ B what is in exactly one. The Jaccard index |A ∩ B| ÷ |A ∪ B| scores the overlap from 0 for disjoint sets to 1 for identical ones.

Comparing customer or email lists, finding shared tags, checking what one version of a list adds or removes, and discrete-mathematics homework are typical uses. The default sets {1, 2, 3, 4, 5} and {4, 5, 6, 7} share 2 of their 7 distinct items, so the Jaccard index is 2/7 ≈ 0.2857.

Items can be numbers or words, separated by commas, semicolons, new lines or spaces. Numbers are compared by value, so 1 and 1.0 are one item; words must match exactly, including capital letters. Results list numbers in order, then words alphabetically.

## Inputs

- **Set A**: Separate items with commas, semicolons or new lines (or spaces if you use none of those). Numbers like 1 and 1.0 count as the same item.
- **Set B**

## Results

- Union A ∪ B — main result
- Intersection A ∩ B
- Difference A − B
- Difference B − A
- Symmetric difference A △ B
- |A|
- |B|
- |A ∪ B|
- |A ∩ B|
- Jaccard index |A ∩ B| / |A ∪ B|

## Formula

$$
\begin{gathered} A \cup B = \{x : x \in A \lor x \in B\} \\[6pt] A \cap B = \{x : x \in A \land x \in B\} \\[6pt] A \mathbin{\triangle} B = (A \setminus B) \cup (B \setminus A) \end{gathered}
$$

## Worked examples

### {1, 2, 3, 4, 5} and {4, 5, 6, 7}

- Set A: 1, 2, 3, 4, 5
- Set B: 4, 5, 6, 7
- **Union A ∪ B: {1, 2, 3, 4, 5, 6, 7}**
- **Intersection A ∩ B: {4, 5}**
- **Difference A − B: {1, 2, 3}**
- **Difference B − A: {6, 7}**
- **Symmetric difference A △ B: {1, 2, 3, 6, 7}**
- **|A ∪ B|: 7**
- **Jaccard index |A ∩ B| / |A ∪ B|: 0.285714**
- Checked against: Python set operators |, &, -, ^ on {1,2,3,4,5} and {4,5,6,7}; 2/7 = 0.285714

### Words with one item in common

- Set A: apple, banana, cherry
- Set B: banana, date
- **Union A ∪ B: {apple, banana, cherry, date}**
- **Intersection A ∩ B: {banana}**
- **Symmetric difference A △ B: {apple, cherry, date}**
- **Jaccard index |A ∩ B| / |A ∪ B|: 0.25**
- Checked against: Python set operators on string sets, sorted()

### Disjoint sets (edge case)

- Set A: 1, 3, 5
- Set B: 2, 4, 6
- **Intersection A ∩ B: ∅**
- **|A ∩ B|: 0**
- **Union A ∪ B: {1, 2, 3, 4, 5, 6}**
- **Jaccard index |A ∩ B| / |A ∪ B|: 0**
- Checked against: Python: {1,3,5} & {2,4,6} == set()

### Equal numbers written differently

- Set A: 1, 1.0, 2
- Set B: 2.00, 3
- **|A|: 2**
- **Intersection A ∩ B: {2}**
- **Union A ∪ B: {1, 2, 3}**
- Checked against: Python: {Decimal('1'), Decimal('1.0'), Decimal('2')} has 2 elements; Decimal('2.00') == Decimal('2')

### Empty set A (edge case)

- Set B: x, y
- **Union A ∪ B: {x, y}**
- **Intersection A ∩ B: ∅**
- **Difference A − B: ∅**
- **Difference B − A: {x, y}**
- **|A|: 0**
- **Jaccard index |A ∩ B| / |A ∪ B|: 0**
- Checked against: Set identities: ∅ ∪ B = B, ∅ ∩ B = ∅, B − ∅ = B

### Space-separated, numbers sorted before words

- Set A: b 10 a 2
- Set B: 10 c
- **Union A ∪ B: {2, 10, a, b, c}**
- **Intersection A ∩ B: {10}**
- **Difference A − B: {2, a, b}**
- Checked against: Python: sorted(numbers) + sorted(words) of the union {2, 10, 'a', 'b', 'c'}

## Questions

### What is the difference between union and intersection?

The union A ∪ B contains every element in A, in B or in both; the intersection A ∩ B contains only the elements in both. For A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7}, the union is {1, 2, 3, 4, 5, 6, 7} and the intersection is {4, 5}. The union is never smaller than either set, and the intersection never larger.

### What is the symmetric difference of two sets?

The elements that are in exactly one of the two sets: A △ B = (A − B) ∪ (B − A), which is the union with the intersection removed. For {1, 2, 3, 4, 5} and {4, 5, 6, 7} it is {1, 2, 3, 6, 7}. It behaves like XOR in logic, and Python's ^ operator computes it for sets.

### How do you find the number of elements in a union?

Use the inclusion–exclusion principle: |A ∪ B| = |A| + |B| − |A ∩ B|, subtracting the intersection because its elements were counted twice. With |A| = 5, |B| = 4 and |A ∩ B| = 2, the union has 5 + 4 − 2 = 7 elements. For three sets it extends to |A| + |B| + |C| − |A ∩ B| − |A ∩ C| − |B ∩ C| + |A ∩ B ∩ C|.

### What is the Jaccard index?

A similarity score for two sets: the size of their intersection divided by the size of their union, from 0 for disjoint sets to 1 for identical ones. {1, 2, 3, 4, 5} and {4, 5, 6, 7} share 2 of 7 distinct items, a Jaccard index of 2/7 ≈ 0.2857. It is used to compare documents, tag sets and search results, and 1 minus the index is the Jaccard distance.

### Is A − B the same as B − A?

No. A − B, also written A \ B, keeps the elements of A that are not in B, while B − A keeps the elements of B that are not in A. With A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7}, A − B = {1, 2, 3} but B − A = {6, 7}. The two differences never share an element, and together they form the symmetric difference.

### How accurate is the set operations calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 6 worked examples whose answers come from independent sources; for example, “{1, 2, 3, 4, 5} and {4, 5, 6, 7}” is checked against Python set operators |, &, -, ^ on {1,2,3,4,5} and {4,5,6,7}; 2/7 = 0.285714.

### Where does the method come from?

Wolfram MathWorld — Union, Intersection, Set Difference, Symmetric Difference; Python documentation — set types and operations.

## Sources

- [Wolfram MathWorld — Union, Intersection, Set Difference, Symmetric Difference](https://mathworld.wolfram.com/SymmetricDifference.html)
- [Python documentation — set types and operations](https://docs.python.org/3/library/stdtypes.html#set-types-set-frozenset)