Union of sets – Venn diagram

In this chapter we will learn to represent union of sets using Venn diagram with examples.

Let’s first review the union operation in set theory.

What is Union in set theory ?

If A & B are two sets, then union operation will result in element present in both set A & B.

For Example;
If A = { 3, 5, 7 }
B = {2, 4, 7, 8 }

Then, A union B = {2, 3, 4, 5, 7, 8}

Symbol of Union of Set

The union operation is represented by symbol ” ∪ “.

So, if we want to perform union of set A & B, we will represent as ” A ∪ B “

Union of Set – General Expression

In set builder form, the union of set A & B can be represented as follows;

A ∪ B = { x : x 𝜖 A or x 𝜖 B }

It says that A union B consist of entity x, where x belongs to either set A or B.

Using Venn diagram in Union Operation

If you want to understand the basics of Venn diagram in set theory, please click the link.

You can use Venn diagram to graphically represent union operation between two or more sets.

If A & B are two sets, the Venn diagram diagram representation is shown below.

Representing union of set through Venn diagram

In the above image;
(a) Rectangular box represent universal set.

(b) Circle A & B represent the set A & B respectively.

(c) The green color represent the union of sets which covers all the element present both in set A & B.

i hope you understood the basic concept, let us now look at the properties of union of set with Venn diagram.

Important Union Property & Operation with Venn diagram

(01) If A ⊆ B, then A ⋃ B = B

It says that if A is subset of B, then union of set A & B will result in set B.

Venn diagram representing union of set

In the above image;

⟹ rectangle represents the universal set

⟹ big circle represent set B and small circle represent set A.

Since A is subset of B, all the elements of set A are present in set B. That’s why circle A is present inside circle B.

⟹ The union of A ⋃ B results in set B which is shown by green color.

(02) A ∪ B = B ∪ A  

In the union operation, if we change the order of set, we will get the same result.

A union B Venn diagram

(03) A ⋃ A’ = Universal set    

The union of set A with its complement will result in universal set.

Venn diagram union with image

In the above image;
⟹ Set A is represented by orange color

⟹ Set A’ (Set A compliment) is represented by pink color

⟹ the union of A ⋃ A’ results in complete rectangle box shown by blue color

(04) A ⋃ B’

Here we are doing union of set A with B complement.

Union of sets Venn diagram

In the first Image;
⟹ Set A is shown by green color
⟹ Set B’ is shown by pink color

After taking union of A ⋃ B’ we get the second image.

Here the blue color represent the solution A ⋃ B’

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