Union of Sets – Set Operations

In this tutorial we will learn, properties of operation of union of sets.

Meaning of operations on sets

There are many operations in maths like division, multiplication, addition, subtraction etc.
When we apply these operations on numbers, we get a new number.

Example: (a) If we multiply 4 by 5, we get a new number as 20.

(b) If we devide 40 by 10, we get a new number as 4.

(c) If we add 21 by 32, we get a new number as 53.

(d) If we subtract 31 by 45, we get a new number as 14.
Similarly, when we apply some operations on sets we get a new set.

First we understand the meaning of Sets, in mathematics and then we learn about operations on sets. 

What is a set?

A collection of different objects having some common properties is called a set. The collection could be anything like alphabets, numbers, pictures etc and all objects are elements of the set.

In mathematics, sets are collections of different things like flowers, numbers, cups, plates, shapes, animals etc.

Examples:  (a) Set of flowers
                  (b) Set of numbers
                  (c) Set of shapes
                  (d) Set of cups
                  (e) Set of animals

The different objects of a set are called the elements of a set. Elements or members of a set are same thing. Generally, we denote a set with capital letters.

First we have to list all objects and separate them with comma and then enclose them in curly braces.

The set of natural numbers, form 2 to 6 is present as
                      A = {2, 3, 4, 5, 6} 

set of even numbers form 4 to 10
                     B = {4, 6, 8, 10}

When we apply some operations on sets we get a new set. 

Definition of Operation of a set

An Operation of a set, is where two are more sets combined, and form one set under the given conditions. 

There are five basic operations on sets.

1. Union of sets

2. Intersection of sets

3. Difference of sets

4. Complement of the sets

5. Cartesian product of sets.  

                       Union of sets                   

Definition: 

The Union of two sets A and B is the smallest set, which contains all the elements of both sets and taking every element of both sets A and B, without repeating any element, or common elements being taken only once.

The symbol ‘∪’ used for the Union of two sets.

Symbolically, we write union of two sets A and B is

A∪B, and read as ‘A union B’.

A∪B = {x: x∈A or x∈B}

Example: If A = {3, 4, 5, 7} and B = {1, 2, 3, 6}. 

then union of set A and set B.

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

The union of two sets can be represented with venn diagram.

                                                   A∪B

 Properties of the Operation of Union

1. A ∪ B = B ∪ A   (Commutative law)

Commutative law says that A∪B and its opposite B∪A will always be equal. 

2. (A ∪ B) ∪ C = A ∪ (B ∪ C)   (Associative law)

Here, order of operation does not have any relevance, the result will always be equal.

3. A ∪ ∅ = A   (Law of Identity) element, ∅ is the identity of ∪)

The union of a set and the empty set will always be the set as empty set, and contains not any element.

4. A ∪ A = A   (Idempotent law)

The union of a set with itself will always be the set itself and we do not consider the repetition of elements.

5. U ∪ A = U   (Law of domination, U, U is the Universal set.)

The union of the set and its universal set will always be the universal set and universal set contains also all the elements of the set.

6. A ∪ A’ = U (Negation)

The union of the set and its complement will always be the universal set, both sets are the subset of universal set and together they contain all the elements of universal set.

Let us see these laws,

1. Commutative law A ∪ B = B ∪ A 

let Universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, 

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

   A ∪ B = {1, 2, 3} ∪ {3, 4, 5}

   A ∪ B = {1, 2, 3} ∪ {3, 4, 5}

 ∴   A ∪ B = {1, 2, 3, 4, 5} 

    B ∪ A = {3, 4, 5,} ∪ {1, 2, 3}

 ∴  B ∪ A = {3, 4, 5, 1, 2, 3}

    B ∪ A = {1, 2, 3, 4, 5} and 

     A ∪ B = {1, 2, 3, 4, 5} 

     ∴   A ∪ B = B ∪ A

2. Associative law

   (A ∪ B) ∪ C = A ∪ (B ∪ C)

A ∪ B =  {1, 2, 3, 4, 5} and (B ∪ C) = {3, 4, 5, 6, 7}

(A ∪ B) ∪ C =  {1, 2, 3, 4, 5} ∪ {5, 6, 7}

(A ∪ B) ∪ C =  {1, 2, 3, 4, 5, 6, 7}

A ∪ (B ∪ C) =  {1, 2, 3,} ∪ {3, 4, 5, 6, 7}

A ∪ (B ∪ C) =  {1, 2, 3, 4, 5, 6, 7} and

(A ∪ B) ∪ C =  {1, 2, 3, 4, 5, 6, 7}

3. Law of Identity (element, ∅ is the identity of ∪)

    A ∪ ∅ = A

We know that empty set have no any element, so the union will be only all the elements of set A.

union will be,  A ∪ ∅ = {1, 2, 3,} ∪ {}

                      A ∪ ∅ = {1, 2, 3}

               ∴  A ∪ ∅ = A

4. Idempotent law

A ∪ A = A                               

               A ∪ A = {1, 2, 3} ∪ {1, 2, 3}

               A ∪ A = {1, 2, 3}

           ∴  A ∪ A = A 

5. Law of Domination, (U), U is the Universal set

U ∪ A = U

Universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, 

set A = {1, 2, 3,}

Union will be all the elements of universal set and the elements of set A, but universal set has all the elements so union will be

U ∪ A = {1, 2, 3, 4, 5, 6, 7, 8, 9} ∪ {1, 2, 3,}

U ∪ A = {1, 2, 3, 4, 5, 6, 7, 8, 9}

  ∴ U ∪ A = U

6. A ∪ A’ = U (Negation)

Universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, 

If set A = {1, 2, 3,} and set A’ = {4, 5, 6, 7, 8, 9}

∴     A ∪ A’ = {1, 2, 3, 4, 5, 6, 7, 8, 9} = U