# Union of Sets

For two sets *A* and *B*, the union of *A* and *B* (denoted by *A∪B*) is the set of all distinct elements that belong to set *A* or set *B*. In mathematical form,

A∪B = { x: x∈A or x∈B}

Similarly, for three sets *A*, *B* and *C*,

A∪B∪C = { x: x∈A or x∈B or x∈C}

## Examples of Union of Sets

### Example #1: Union of Two Sets With Venn Diagram

**If A = {a, b, c, d, e} and B = {d, e, f, g}, find A∪B. **

Here, A∪B = {a, b, c, d, e} ∪ {d, e, f, g} ∴ A∪B = {a, b, c, d, e, f, g}

### Example #2

**Suppose A = { x: x is an integer between 2 and 7} and B = { x: x is an integer between 5 and 10} then, find A∪B.**

Here, A = {3, 4, 5, 6} B = {6, 7, 8, 9} ∴ A∪B = {3, 4, 5, 6, 7, 8, 9} = { x: x is an integer between 2 and 10}

### Example #3: Union of Disjoint Set With Venn Diagram

**If A = {a, b, c} and B = {d, e, f, g}, find A∪B.**

Here, A∪B = {a, b, c} ∪ {d, e, f, g} ∴ A∪B = {a, b, c, d, e, f, g}

### Example #4

**Suppose A = { x: x is an odd integer} and B = { x: x is an even integer} then, find A∪B.**

Here, A = {..., -5, -3, -1, 1, 3, 5, ...} B = {..., -4, -2, 0, 2, 4, 6, ...} ∴ A∪B = {..., -3, -2, -1, 0, 1, 2, ... } = {x: x is an integer}

### Example #5: Union of Three Sets With Venn Diagram

**If A = {a, b, c, d, e}, B = {d, e, f, g} and C = {c, e, f, h, i}, find A∪B∪C.**

Here, A∪B∪C = {a, b, c, d, e} ∪ {d, e, f, g} ∪ {c, e, f, h, i} ∴ A∪B = {a, b, c, d, e, f, g, h, i}

## Properties of Union of Set

### Commutative Property

If *A* and *B* are two sets then, **A∪B = B∪A**

Suppose, A = {1, 2, 3} B = {3, 4, 5} A∪B = {1, 2, 3, 4, 5} B∪A = {3, 4, 5, 1, 2} ∴ A∪B = B∪A

### Associative Property

If *A*, *B* and *C* are three sets then*, ***A∪(B∪C)= (A∪B)∪C **

Suppose, A = {1, 2, 3} B = {3, 4, 5} C = {3, 1, 5, 8, 9} B∪C = {3, 4, 5, 1, 8, 9} A∪B = {1, 2, 3, 4, 5} A∪(B∪C) = {1, 2, 3, 4, 5, 8,9} (A∪B)∪C = {1, 2, 3, 4, 5, 8, 9} ∴ A∪(B∪C) = (A∪B)∪C

### Identity Property

The union of a set and the empty set it that set itself, i.e, A∪ϕ = A.

Suppose, A = {1, 2, 3} B = ϕ = {} A∪B = {1, 2, 3, } ∪ {} = {1, 2, ,3} ∴ A∪ϕ = A