In these lessons, we will learn the union of sets and the complement of the union of sets. For more lessons, see our collection of lessons on sets.

**Related Pages**

Union Of Sets

Intersection Of Two Sets

Venn Diagrams

More Lessons On Sets

More Lessons for GCSE Maths

Math Worksheets

The **union** of two sets A and B is the set of elements, which are in
A **or** in B **or** in both. It is denoted by A ∪ B and is read ‘**A union B**’.

The following table gives some properties of Union of Sets: Commutative, Associative, Identity and Distributive. Scroll down the page for more examples.

**Example:**

Given U = {1, 2, 3, 4, 5, 6, 7, 8, 10}

X = {1, 2, 6, 7} and Y = {1, 3, 4, 5, 8}

Find X ∪ Y and draw a Venn diagram to illustrate X ∪ Y.

**Solution:**

X ∪ Y = {1, 2, 3, 4, 5, 6, 7, 8} ← 1 is written only once.

If **X ⊂ Y** then **X ∪ Y = Y.**

We will illustrate this relationship in the following example.

**Example:**

Given U = {1, 2, 3, 4, 5, 6, 7, 8, 10}

X = {1, 6, 9} and Y = {1, 3, 5, 6, 8, 9}

Find X ∪ Y and draw a Venn diagram to illustrate X ∪ Y.

**Solution:**

X ∪ Y = {1, 3, 5, 6, 8, 9}

The **complement of the set X ∪ Y** is the set of elements that are members of the universal set U
but are not in X ∪Y. It is denoted by (X ∪ Y)’

**Example:**

Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9}

X = {1, 2, 6, 7} and Y = {1, 3, 4, 5, 8}

a) Draw a Venn diagram to illustrate ( X ∪ Y ) ’

b) Find ( X ∪ Y ) ’

**Solution:**

a) First, fill in the elements for X ∩ Y = {1}

Fill in the other elements for X and Y and for U

Shade the region outside X ∪ Y to indicate (X ∪ Y)’

b) We can see from the Venn diagram that

(X ∪ Y)’ = {9}

Or we find that X ∪ Y = {1, 2, 3, 4, 5, 6, 7, 8} and so

(X ∪ Y)’ = {9}

**Example:**

Given U = {x : 1 ≤ x ≤10, x is an integer}, A = The set of odd numbers, B = The set of factors of
24 and C = {3, 10}.

a) Draw a Venn diagram to show the relationship.

b) Using the Venn diagram or otherwise, find:

i) (A ∪ B ) ’ ii) (A ∪ C ) ’ iii) (A ∪ B ∪ C ) ’

**Solution:**

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

a) First, fill in the elements for A ∩ B ∩C = {3}, A ∩ B {1, 3},

A ∩ C = {3}, B ∩ C = {3} and then the other elements.

b) We can see from the Venn diagram that

i) (A ∪ B ) ’ = {10}

ii) (A ∪ C ) ’ = {2, 4, 6, 8}

iii) (A ∪ B ∪ C ) ’ = { }

∪ is the union symbol and can be read as “or”. The union of two sets are all the elements form both sets.

∩ is the intersection symbol and can be read as “and”. The intersection of two sets are those elements that belong to both sets.

The **intersection** of two sets are those elements that belong to both sets.

The **union** of two sets are all the elements from both sets.

**Example:**

If D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} with subsets A, B and C where A = {4, 6, 8} and B = {6, 7, 8, 9}
and C = {1, 2, 3, 4}, find the following:

A ∩ B

B ∩ C

A ∪ B

B ∪ C

(A ∪ B ∪ C)'

This video shows how to shade the union, intersection and complement of two sets.

**Example:**

Shade the indicated region

(1) A ∪ B'

(2) A' ∩ B'

(3) (A ∪ B)'

Try the free Mathway calculator and
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