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.

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
problem solver below to practice various math topics. Try the given examples, or type in your own
problem and check your answer with the step-by-step explanations.

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