# Set intersection calculator

World's simplest set tool
With this online application, you can quickly calculate the intersection of multiple sets and find the common elements among them. You can also intersect multisets with duplicate elements. As there are various set notations, you can change the format of the input sets as well as create a new format for the output set. Optionally, you can sort the elements of the intersection in alphabetical or alphanumerical order. Created by team Browserling.

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## What is a set intersection calculator?

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This browser-based program finds the intersection of two or more sets. The intersection of two sets A and B is a set C, such that its elements belong to both set A and set B. Similarly, the intersection of three, four, or more sets is a set that consists of common elements of all the given sets. Set intersection is a commutative binary operation and is denoted by the symbol ∩. Formally, the mathematical definition of intersection of two sets A and B is as follows: A∩B = {x: x ∈ A and x ∈ B}. For example, if the set A is {2, 4, 5, 9} and the set B is {4, 7, 8, 9}, then their intersection is the set {4, 9} (because 4 and 9 belong to both sets). If one of the sets is a subset of the other set, A ⊆ B, then A∩B = A. For example, if A = {a, b} and B = {a, b, c, d}, then A∩B = {a, b} = A. When computing the intersection operation on disjoint sets A and B (sets are disjoint if they don't share common elements), the result is the empty set with no elements in it. For example, sets {a, b} and {1, 2} are disjoint, therefore {a, b}∩{1, 2} is ∅. In the input field of this application, you can enter as many sets as you need. To indicate where one set ends and another set begins, use a delimiter string of three dashes "---". By default, this delimiter string is preconfigured in the options but you can change it to any other string that suits your needs. You can also adjust the format of the sets by changing the set braces { } to other symbols and changing the comma separator of the elements to a different separator. It's important that all input sets adhere to the same set format as configured in the options. The output set that contains the intersection can be also customized and you can specify a new set format using similar options. In the classic set theory, sets such as {a, a, a, b} and {a, b} are equal because the repeated element copies can be ignored. In our utility, we also added support for multisets. If the option "Multiset Intersection" is selected, then the repeating elements aren't ignored. With this option turned on, {a, a, b, c}∩{f, a, c, a} = {a, a, c}. Usually, the order of elements in a set isn't important, but if you need to, you can sort the elements in alphabetical or numerical order. Setabulous!

## Set intersection calculator examples

Click to useFind Common Divisors

In this example, we use the intersect operator to find the common factors of two numbers. The first set consists of all divisors of the number 12 and the second set consists of all divisors of the number 16. The intersection of these two sets contains common divisors of both integers. Both input sets use the classical set format (comma-separated elements enclosed in curly brackets). The sets themselves are separated by a line of three dashes. We preserve the input set format in the output set and find that the three common factors of integers 12 and 16 are 1, 2, and 4.

{1, 2, 3, 4, 6, 12}
---
{1, 2, 4, 8, 16}

{1, 2, 4}

**Required options**

String that delimits input sets.
(Three dashes by default.)

Symbol that delimits input set
elements. (Comma by default.)

Open set symbol.

Close set symbol.

Repeated elements are not
ignored when calculating
intersection of sets.

Symbol that delimits output set
elements. (Comma by default.)

Open set symbol.

Close set symbol.

Don't sort the elements in
the output set.

Find Common Letters

The input sets in this example are words enclosed in quotation marks. As funny as it may sound, the quote characters can be viewed as open and close set symbols, the letters can be viewed as set elements, and the nothing symbol (empty symbol) can be viewed as set element separator. The quotation marks indicate the beginning and end of the set, so we put them in the open-set and close-set option fields. Each letter of the word is an element of the set, so we leave the set element separator option field empty. Each word is located on a new line, so we use the newline character as individual set separator. Now when we find the intersections of these sets, it turns out the three words share only four letters. The shared letters are sorted in the output and printed as a new set in a new format with single quotes as set symbols and the space character as set element separator.

"hamster"
"rhinoceros"
"horse"

'e h r s'

**Required options**

String that delimits input sets.
(Three dashes by default.)

Symbol that delimits input set
elements. (Comma by default.)

Open set symbol.

Close set symbol.

Repeated elements are not
ignored when calculating
intersection of sets.

Symbol that delimits output set
elements. (Comma by default.)

Open set symbol.

Close set symbol.

Sort the elements of the set
intersection alphabetically.

Multiset Intersection

In this example, we activate the "Multiset Intersection" option, which means that the intersection of sets takes into account the number of copies of repeated elements. The input sets use square brackets and are separated by the ∩ symbol. To correctly parse the input, we enter the ∩ character in the set separator field and to remove the square brackets, we type them in the set-open and set-close options, and we put a comma in the item separator field. Once the multiset intersection has been calculated, the elements are sorted in alphanumerical order. They are then separated with a semicolon, enclosed in regular parenthesis, and printed in the output area.

[x, x, y, 5, 10, -1, 3, 10] ∩ [z, x, x, 5, 10, -2, 5, 10] ∩ [x, i, x, -1, 7, 5, 10, 10]

(5; 10; 10; x; x)

**Required options**

String that delimits input sets.
(Three dashes by default.)

Symbol that delimits input set
elements. (Comma by default.)

Open set symbol.

Close set symbol.

Repeated elements are not
ignored when calculating
intersection of sets.

Symbol that delimits output set
elements. (Comma by default.)

Open set symbol.

Close set symbol.

Sort the elements of the set
intersection alphabetically
and numerically.

Pro tips
Master online set tools

You can pass input to this tool via

__?input__query argument and it will automatically compute output. Here's how to type it in your browser's address bar. Click to try!
https://onlinesettools.com/find-set-intersection

__?input__=%7B1%2C%202%2C%203%2C%204%2C%206%2C%2012%7D%0A---%0A%7B1%2C%202%2C%204%2C%208%2C%2016%7D&input-set-separator=---&input-element-separator=%2C&input-open-set=%7B&input-close-set=%7D&allow-repetitions=false&output-element-separator=%2C%20&output-open-set=%7B&output-close-set=%7D&do-not-sort=true
All set tools

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Draw a Venn Diagram

Illustrate two or more sets as a Venn diagram.

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Apply the set difference operation on sets A and B.

Find Set Cartesian Product

Apply the set cartesian product operation on sets A and B.

Find All Subsets of a Set

Quickly find all sets that are subsets of set A.

Find All Set Permutations

Generate all permutations of set elements.

Enumerate a Set

Add numbering to all set elements.

Filter a Set

Print set elements that match a filter.

Find Set Elements

Find elements in a set that match certain criteria.

Apply a Function on a Set

Run a function on all elements in a set.

Convert a Multiset to a Set

Convert a set with repeated elements to a standard set.

Convert a Set to a Multiset

Convert a standard set to a multiset with repeated elements.

Convert a Set to a List

Create a list from the given set.

Convert a List to a Set

Create a set from the given list.

Convert a Set to an Array

Create an array from the given set.

Convert an Array to a Set

Create a set from the given array.

Duplicate Set Elements

Repeat set elements multiple times.

Print Duplicate Set Elements

Find all duplicate elements in a set.

Remove Duplicate Set Elements

Delete all duplicate elements from a set (leave unique).

Print Unique Set Elements

Find all unique elements in a set.

Remove Unique Set Elements

Delete all unique elements from a set (leave duplicates).

Remove Empty Set Elements

Delete empty elements (zero-length elements) from a set.

Find Set Depth

Calculate how many levels of subsets a set has.

Flatten a Set

Decrease subset nesting.

Truncate a Set

Remove elements from a set and make it smaller.

Truncate Set Elements

Shorten all set elements to the given length.

Expand a Set

Add elements to a set and make it bigger.

Split a Set

Split a set into a certain number of subsets.

Join Sets

Merge multiple sets together to form one large set.

Slice a Set

Extract an index-based subset from a set.

Partition a Set

Find disjoint subsets of the given set whose union is the same set.

Randomize a Set

Randomly change the order of elements in a set.

Select a Random Set Element

Pick a random element from the given set.

Select a Random Subset

Pick a random subset of the given set.

Generate an Empty Set

Create a set with no elements.

Generate a Digit Set

Create a set that contains digits.

Generate a Number Set

Create a set that contains numbers.

Generate an Integer Set

Create a set that contains integers.

Generate a Decimal Set

Create a set that contains decimal fractions.

Generate a Letter Set

Create a set that contains letters.

Generate a Character Set

Create a set that contains characters.

Generate a Word Set

Create a set that contains words.

Generate a String Set

Create a set that contains strings.

Generate a Text Set

Create a set that contains text.

Generate a Sentence Set

Create a set that contains sentences.

Generate a Random Set

Create a set that contains random elements.

Generate a Custom Set

Create a custom set with custom elements and custom size.

Generate an Infinite Set

Create a set with infinitely many elements.

Generate a Finite Set

Create a set with a finite number of elements.

Change Set Notation

Change the open-set, close-set, and element separator symbols.

Change Set Size

Add or remove set elements to make it a certain size/length.

Destroy a Set

Launch a Zalgo attack on a set and destroy it.

Compare Sets

Find all differences between two or more sets.

Symmetrize a Set

Convert a regular set to a symmetric multi-set.

Color a Set

Add colors to set elements.

Visualize a Set

Create an abstract visualization of a set.

Convert a Set to an Image

Create a downloadable picture from a set.

Print Set Analytics

Analyze a set and print its statistics.

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