Union of sets

This lesson will explain how to find the union of sets. We will start with a definition of the union of two sets.

Definition: Given two sets A and B, the union is the set that contains elements or objects that belong to either A or to B or to both.

We write A ∪ B


Basically, we find A ∪ B by putting all the elements of A and B together. We next illustrate with examples.


Union of 2 sets
Example #1.


Let A = {1 orange, 1 pineapple, 1 banana, 1 apple} and B = { 1 spoon, 1 knife, 1 fork}

A ∪ B = {1 orange, 1 pineapple, 1 banana, 1 apple, 1 spoon, 1 knife, 1 fork}

Example #2.

Find the union of A and B

Union of sets A and B


A = { 1, 2, 4, 6} and B = { 4, a, b, c, d, f}

A ∪ B = { 1, 2, 4, 6, 4, a, b, c, d, f} = { 1, 2, 4, 6, a, b, c, d, f }

Notice that it is perfectly OK to write 4 once or twice.

Example #3.

A = { x / x is a number bigger than 4 and smaller than 8}

B = { x / x is a positive number smaller than 7}

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

A ∪ B = { 1, 2, 3, 4, 5, 6, 7}

Or A ∪ B = { x / x is a number bigger than 0 and smaller than 8}

Again, notice that 5 and 6 were written only once although it would be perfectly OK to write them twice.

Example #4.

A = { x / x is a country in Asia}

B = { x / x is a country in Africa}

A ∪ B = {x / x is a country in Asia and Africa} = { all countries in Asia and Africa}

Technically, using the definition, we should have said A ∪ B = {x / x is a country in either Asia or Africa or both countries}

However, in real life, this is not the way we talk. By saying all countries in Asia and Africa, it is usually understood in daily conversations that we are talking about all countries in Asia and all countries in Africa.

Example #5.

Union of disjoint sets
A = {#, %, &, * , $ }

B = { }

A ∪ B = {#, %, &,* , $}


You may have definitely noticed that the union of sets is simply found by putting the elements of the sets together, preferably without repetition.

Definition of the union of three sets

Given three sets A, B, and C the union is the set that contains elements or objects that belong to either A, B, or to C or to all three.

We write A ∪ B ∪ C

Basically, we find A ∪ B ∪ C by putting all the elements of A, B, and C together.

A = { 1, 2, 4, 6}, B = { a, b, c,} and C = A = {#, %, &, * , $ }

A ∪ B ∪ C = { 1, 2, 4, 6, a, b, c,#, %, &, * , $ }

The first graph in the beginning of this lesson shows the shaded region for the union of two sets.

The graph below shows the shaded region for the union of three sets

Union of 3 sets


This ends the lesson about union of sets. If you have any questions about the union of sets, I will be more than happy to answer them.

Use the quiz below to see how well you can find the union of sets

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