# Union

In set theory, the union (∪) of a collection of sets is the set that contains all of the elements in the collection. For example, given two sets, A = {2, 2, 4, 6, 8, 10} and B = {1, 3, 5, 7, 9}, their union is as follows:

A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9 10}

Notice that even though A has two 2s, there is only one 2 in A ∪ B. This is because the union operation includes only one of each unique element.

The union of two sets is commonly depicted using a Venn diagram, in which a set is represented by a circle. The Venn diagram below shows two sets: A = {a, b, c, d, e} and B = {d, e, f, g}. Their union is the set {a, b, c, d, e, f, g} and is represented in the Venn diagram by the area covered by both circles.

## Properties of unions

Like other basic operations such as addition, set operations like unions also have certain properties. Refer to the set page if necessary for a table of symbols commonly used in set theory.

### Unions and subsets

If set A is a subset of set B, then the union of the two sets is set B. Using set notation:

if A ⊆ B, then A ∪ B = B

For example, if A = {2n|n ∈ ℕ} and B is the set of integers, then A ∪ B = B, since set A is the set of positive even integers, which is a subset of all integers.

### Commutative law

The commutative law states that the order in which the union of two sets is taken does not matter. Given two sets, A and B:

A ∪ B = B ∪ A

Let A = {1, 2, 3} and B = {3, 5, 7}. The sets share 1 common element. The union of the sets includes all unique elements of both sets. Thus, A ∪ B = B ∪ A = {1, 2, 3, 5, 7}. Regardless whether A or B is considered first, the result is the same. If B's elements were written first, the union of A and B could be written as {3, 5, 7, 1, 2}. The order in which elements are listed in a set does not matter; the number of elements and the values of the elements determine the set, so the above 2 sets are equal, as are any sets including all the same elements written in different orders.

### Associative law

The associative law states that rearranging the parentheses in a union of sets does not change the result. Given sets A, B, and C:

(A ∪ B) ∪ C = A ∪ (B ∪ C)

### Distributive law

For sets A, B, and C, the distributive law states

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

where A is distributed to B and C. This is similar to the distributive property of multiplication in which multiplication distributes over addition.

Example

Let A = {4, 6, 8, 10}, B = {8, 9, 10, 11}, and C = {10, 11, 12}. Show that A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).

Compute the left side of the equation:

B ∩ C = {10, 11}

A ∪ (B ∩ C) = {4, 6, 8, 10} ∪ {10, 11} = {4, 6, 8, 10, 11}

Compute the right side of the equation:

A ∪ B = {4, 6, 8, 9, 10, 11}

A ∪ C = {4, 6, 8, 10, 11, 12}

(A ∪ B) ∩ (A ∪ C) = {4, 6, 8, 10, 11}

In both cases, the resulting set is {4, 6, 8, 10, 11}.

### De Morgan's laws

In set theory, De Morgan's laws are a set of rules that relate the union and intersection of sets through their complements.

#### Union of sets:

The complement of the union of two sets is equal to the intersection of their complements:

(A ∪ B)C = AC ∩ BC

Given that A and B are subsets of the universal set 𝕌, this relationship can be seen in the figure below:

The union of A and B, A ∪ B, is shaded in blue. Its complement, (A ∪ B)C is shaded in yellow. The intersection of the complements of A and B, AC ∩ BC is also shaded in yellow.

#### Intersection of sets:

The complement of the intersection of two sets is equal to the union of their complements:

A ∩ B = AC ∪ BC

Given that A and B are subsets of the universal set 𝕌, this relationship can be seen in the figure below:

The intersection of A and B, A ∩ B, is shaded in red. Its complement, (A ∩ B)C is shaded in grey. The union of the complements of A and B, AC ∪ BC, is also shaded in grey.