Operation on Sets Intersection of Sets and Difference of Two Sets

Set Operations

In our everyday life, we deal with collections of objects (person, numbers or any other thing).

For example, consider the following collections:

i) Collection of all the students in your class.

ii) Collection of all the teachers at your school. 

iii) Collection of all counting numbers that are less than 10 i.e., of numbers 1, 2, 3, 4, 5, 6, 7, 8, 9.

iv) Collection of all even natural numbers less than 15 i.e., of numbers 2, 4, 6, 8, 10, 12, 14.

v) Collection of the first five natural numbers divisible by 5 i.e., of the numbers 5,10, 15, 20, 25.

vi) Collection of all vowels in the English alphabets i.e., of the letters a, e, i, o, u.

vii) Collection of all the days in a week.

viii) Collection of all the books in your bag. 

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Note: All the above collections is a well-defined collection of objects.

A ‘well-defined collection of objects’ means that if we are provided with a collection and an object, then it would be possible to assert without any doubt that if an object belongs to the collection or not. 


What is the Data Set?

A set is a defined collection of objects

The objects that belong to the set are called its members or elements. Each of the above collections is a set. 

Now consider the following collections:

i) Collection of all the intelligent students in your class. 

We cannot call it a well-defined collection because people may differ on whether a student of your class is intelligent or not.  

ii) Collection of all the competent teachers of your school.

We cannot call it a well-defined collection because people may differ on whether a teacher of your school is competent or not.

iii) Collection of four days of a week.

We cannot call it a well-defined collection because it is not known which four days of a week are to be included in the collection. 

Any of the above collections are not a set.


Notation

The sets are usually denoted by capital letters A, B, C, and so on… The members of a set are denoted by small letters x, y, z, and so on. 

If x is a member of the set A, we write x ∈ A (read as ‘x belongs to A’) and if x is not a member of the set A, we write x ∉ A (read as ‘x does not belong to A’)

If x and y are the members of the set A, we write x, y ∈ A.


Representation of A Set

A set can be represented by the following method:

  1. Description method

  2. Roster method or tabular form

  3. Rule method or set builder form.


Types of Sets

There are Four Types of Sets

Finite Set

Infinite Set

Empty Set

Equal Set

A set that contains a limited (countable) number of different elements is called finite. In other words, a set will be finite if the counting of its different elements comes to an end.

Eg: A = {a, b, c, d,e}

A set that contains an infinite (uncountable) number of different elements is called infinite. In other words, a set is called infinite if the counting of its different elements does not come to an end. 

Eg: N = {1, 2, 3, ....}

A set with no elements is called an empty set. It is also called null or void set. There is only one such set which is denoted as φ or

Eg: {x:x ∈ N and x<1}

Two sets are called equal if they have the same elements.


 Union of Sets

The union of two sets A and B is the set consisting of all the elements which belong to either A or B or both. We write it as A U B.

Example, if A = { a, e, i, o, u } and B = { a, b, c, d, e } the, 

             A U B = { a, e, i, o, u, b, c, d }


Intersection of Sets

The intersection of two sets A and B is the set consisting of all elements which belong to both A and B. we write it as A ∩ B.

Example Of Intersection Of Sets

 i) If A = {a, e, i, o, u} and B = {a, b, c, d, e} then, 

A ∩ B= {a,e}

ii) If A = {the colours of the rainbow} = {Violet, Indigo, Blue, Green, Yellow, Orange, Red}

And B = {Black, Red, Blue, White} the 

A ∩ B = {Red, Blue}


Difference of Sets

The difference of two sets A and B is a set with no elements in common. 

For example, 

i) A = {1, 3, 5, 7, 9} and B = {0, 2, 4, 6, 8, 10}

There is a difference of two sets A and B as there are no common elements between them. 


Properties of Set Operations

Union of Sets 

Properties of (A U B) are:

  1. The commutative law holds true as (A U B) = (B U A)

  2. The associative property too holds true as (A U B) U {C} = {A} U (B U C)

Intersection of Sets

Properties of – A ∩ B are:

  1. Commutative law =  (A ∩ B) ∩ C = A ∩ (B ∩ C)

  2. Associative law =  A ∩ B = B ∩ A 

  3. Distributive law = A ∩ (B U C) = (A ∩ B) U (A ∩ C)

  4. φ ∩ A = φ

  5. U ∩ A = A 

FAQ (Frequently Asked Questions)

1. What is the Difference Between the Roster Method and Rule Method

A roster method is a method where we list all the members of the set and separate these by commas. The list is enclosed in curly brackets. For example, A = {2, 4, 6, 8, 10, 12, 14}. It is also known as tabular form. 

In a Rule method, we write a variable representing any member of the set followed by a property satisfied by each member of the set and enclose it in curly brackets. This method is also known as the builder form. 

Example: If A is the set consisting of elements x having property p, we write A = {x:x has property p} which is read as ‘the set of elements x such that x has property p.

2. What is the Cardinal Number of a Set?

The number of distinct elements in a finite set is called its cardinal number. The cardinal number of a finite set A can be represented as n(A).

Example, A = {a, b, c, d, e} has 5 elements, so n(A) = 5.