In these lessons, we will learn the union of sets and the complement of the union of sets.

Related Topics: More Lessons on Sets

### Union of 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*’

### Complement of the Union of Sets

### Videos

Sets: Union and Intersection.

∪ 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.

A mathematics lesson on set operation of union.

Union of Sets.
The following video describes the Union and Intersection of Sets.

Union, Intersection and Complement.
Venn Diagrams: Shading Regions. This video shows how to shade the union, intersection and complement of two sets.

You can use the free Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

Related Topics: More Lessons on Sets

The

**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** ⊂

**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**

**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* ) ’ = { }

Sets: Union and Intersection.

∪ 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.

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway widget below to practice Algebra or other 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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