# Union of Sets

The union of two or more sets is a set that contains all the elements from the original sets without any repetition. If an element appears in any set, it will also appear in the union.

## Symbol

The union operation is denoted by the symbol â€˜âˆª.â€™

The union of two sets, A and B, is a new set denoted by A âˆª B, which contains all the elements of sets A and B without repetition. It is read as â€˜A union B.â€™

Let us consider an example.

If A = {1, 2, 3, 4, 5} and B = {6, 7, 10}

Then, A âˆª B = {1, 2, 3, 4, 5, 6, 7, 10}

## Formula

The formula for the union of two sets (A union B ) is given in set builder notation as:

A âˆª B = {x | x âˆˆ A or x âˆˆ B}

This means x is an element of A âˆª B if and only if x is an element of A or x is an element of B.

## Union of Sets and Venn Diagram

In the Venn diagram, the shaded region illustrates the union of sets A and B. It covers all the elements from both sets, where U is the universal set.

Here, A âˆª B = {x | x âˆˆ A or x âˆˆ B}

For example,

If set A = {1, 5, 10, 15, 20, 25, 30} and set B = {25, 26, 27, 28, 29, 30}, then the union of these two disjoint sets is

A âˆª B = {1, 5, 10, 15, 20, 25, 26, 27, 28, 29, 30}.

The result is thus a new set formed by combining all elements of the given sets, and its Venn diagram is shown:

## Number of Elements in A âˆª B

If A and B are two sets, we can calculate the number of elements in their union set using the formula:

n(A âˆª B) = n(A) + n(B) – n(A âˆ© B)

Here,

n(A âˆª B) = Cardinality of set  A âˆª B

n(A) = Cardinality of set A

n(B) = Cardinality of set B

## Properties

### Commutative

The union of two or more sets follows the commutative law. Thus, if A and B are two sets, then A âˆª B = B âˆª A

For example,

If A = {a, b} and B = {b, c, d}, then A âˆª B = {a, b, c, d} and B âˆª A = {b, c, d, a}

A âˆª B = B âˆª A

### Associative

The union of three or more sets does not depend on how we group the sets. Thus, if A and B are two sets, then (A âˆª B) âˆª C = A âˆª (B âˆª C)

For example,

If A = {a, b}, B = {b, c, d}, and C = {c, d, e, f}, then

(A âˆª B) âˆª C = {a, b, c, d} âˆª {c, d, e, f} = {a, b, c, d, e, f}

A âˆª (B âˆª C) = {a, b} âˆª {b, c, d, e, f} = {a, b, c, d, e, f}

Thus, the associative law is satisfied.

### Distributive

If A, B, and C are three different sets, then according to the distributive law

### Identity

The union of an empty set and a set A produces the set itself.

A âˆª É¸ = A

For example,

If A = {a, b} and É¸ = { }, then A âˆª É¸ = {a, b} âˆª { } = {a, b} = A

### Idempotent

The union of a set A with itself is the set A.

A âˆª A = A

For example,

If A = {a, b}, then A âˆª A = {a, b} âˆª {a, b} = {a, b} = A

### Domination

The union of a universal set U and its subset A is the universal set itself.

A âˆª U = U

For example,

If A = {a, b} and U = {a, b, p, q, s}, then A âˆª U = {a, b} âˆª {a, b, p, q, s} = {a, b, p, q, s} = U

## Unions and Subsets

If set A is a subset of set B, then the union of the two sets gives set B.

Using set notation,

If Aâ€‰âŠ†â€‰B, then Aâ€‰âˆªâ€‰B = B

For example,

If A = {b} and B = {b, c, d}, then Aâ€‰âˆªâ€‰B = {b, c, d} = B

## Solved Examples

Find the number of elements in the union of two sets, A and B, when n(A) = 6, n(B) = 5, and n(Aâ€‰âˆ©â€‰B) = 2

Solution:

As we know,
n(A âˆª B) = n(A) + n(B) – n(A âˆ© B)
Here,
n(A) = 6
n(B) = 5, and