CHAPTER-1 SETS

Introduction of sets

Properties of sets

Methods of writing sets

Types of sets

Cardinal number of set

MATHEMATICS

Set:The word set means collection of objects

A set is a collection of well defined objects .

• The word well defined means a set has specific properties.

• Example:1) collection of numbers from 1 to 5

So you can say 1,2,3,4,5

It is well defined set.

2) group of smart students of class 6 .

So in this you cannot define or measure. It is a relative term.

It is not a set.

Let’s see more examples: 1) collection of seven wonders of world.

It is a set.2) Vowels in English alphabet

It is a set.3) Strong students in class VI.

It is not a set. 4) Collection of great people in the world.

It is not a set.5) Collection of first five prime numbers.

It is a set

Link :https://youtu.be/l3-A0O42Lyo

How to write a set:

1) Sets are denoted by capital letters.

• A = {a,e,i,o,u}

2)Elements are always written in small letters.

• A={a,e,i ,o,u}

3) Elements in a set are separated by comma.

• A={a,e,i,o,u}

4) Elements are always written in a set.

• A={ a,e,i,o,u}

• Link: https://youtu.be/TTPsXnhevlI

• Elements or members :

The objects in a set are called its elements or members.

Example : A ={ 1,2,3,4,5}

Here 1,2,3,4,5 are called elements or members of set A.

And suppose we take 7,8,9

Then they are not elements or members of set A .

Properties of sets:

1) Change in order of the elements does not change the set.

• Example: A={1,3,5} B ={5,1,3}

Both Sets A and B are same.

• Example: A ={a,b,c} B ={b,c,a }

Both sets A and B are same.

2) If one or more elements in a set is repeated the set does not change.

• Example: A={1,3,5} B={1,3,3,5,1}

Both Sets A and B are same

• Example: A={a,b,c,b,a} B={a,b,c}

. Both sets A and B are same.

Link: https://youtu.be/y-Dg4sWVIgs

3) Each element of the set is listed once.

• Example : Letters in word BOMBAY.X={ B,O,M,A,Y}

• Example: Letters in word AGARTALA.

A={A,G,R,T,L}4) If number of elements in a set is large then the set is represented by writing elements followed by three dots.

• Example : multiples of 2

A ={2,4,6,…}• Example: Set of natural numbers less than 1000

A={1,2,3,…,999}

Ways of representing a set:

Method 1:

Description method: In this well defined description of the elements of the set and enclosed in curly brackets.

Example: 1)The set of all days of a week.

A= {all days of a week}

2)The set of natural numbers less than 10

B = {natural numbers less than 10}

Method 2:

Roster / Tabular method : In this method all elements of the set are listed , separated by comma and enclosed in curly brackets.

Example : 1) The set of vowels in English alphabet.

A = { a,e,i,o,u}

Example : 2)The set of natural numbers less than 10.

B = { 1,2,3,4,5,6,7,8,9

Link :https://youtu.be/LumU80IN748

Method 3:

Rule or Set builder method: In this method a variable (say x) followed by colon ( : )or vertical line ( | ) is written to denote each element of set.

After the sign of colon or vertical line the characteristic property is written and enclosed in curly brackets.

• Example:1) The set of of all days in a week.

A= {x|x is a day in a week}

• Example:2) The set of natural numbers less than 10.

B = {x|x is a natural number less than 10}

| - denotes ‘such that’

Link :https://www.youtube.com/playlist?list=PLmdFyQYShrjfi7EeDyHxr0jhoPXEOlFX0

Let’s see examples : 1) Vowels of English alphabet.

2)All multiples of 3.

Description method. Roster method. Rule or set builder

Example 1

A={all vowels of English A ={ a,e,i,o,u} A= { x|x is a vowel in

alphabet} English alphabet}

Example 2

A= {all multiples of 3}. A = { 3,6,9,…} A= { x|x is a multiple of 3}

Test yourself

Q1 Find whether it is a set or not a set

a) Collection of numbers less than 50.

b)Collection of tall people in your colony.

c) Collection of tasty fruits.

d) Collection of days of the week.

Q2 True or False

a) {a,b,c} = {c,b,a}

b) {1,2,3} = { a,b,c}

c) {a,a,a} = {a}

• Q3 Write in tabular form

a) Set of multiples of 5

b) Set of prime numbers less than 10

Q4 write in set builder form

a) Set of whole numbers less than 20.

b) Set of factors of 40.

Q5 write in descriptive form

a) Set of all states of India.

b) Set of all natural numbers.

Check your answers

Q1) a)A set b) not a set.

c) not a set. d) A set

Q2) a) true. b ) false. c) false

Q3) a) A={ 5,10,15,…}

b) B ={ 2,3,5,7}

Q4) a) A={ x|x is a whole number less than 20}

b) B ={ x|x is a factor of 40}

Q5) a) A ={ all states of India}

b) B ={all natural numbers}

Types of sets:1) Finite set: A set that contains definite or limited number of elements is called a finite set. Elements can be counted.

Example: 1) A={months in a year} 2) B ={all prime numbers less than 20}

The elements in the above sets are countable so they are finite sets.

2) Infinite set : A set that contains unlimited or uncountable number of elements is called an infinite set. Elements cannot be counted.Example : N ={ all natural numbers}

B ={4,8,12,16,…}

The elements in the above sets are uncountable so they are infinite sets.

Link: .https://youtu.be/BydchiZ8t6o

3) Singleton set : A set that contains only one element is called a singleton setExample: 1) A ={ 3}

2) B={ all months having less than 30 days}In the above examples both sets have one element so they are singleton set.

4) Empty set : A set that contains no elements is called empty set. It is also called null set or void set. Example : 1) set of natural numbers between 1 and 2.

A ={ }2) set of prime numbers less than 2P= { }

In the above examples both sets have no elements.So they are empty sets. Symbol of Empty SetEmpty set has no elements so it is called finite set.

Link: https://youtu.be/OAAwCIndLxA

5) Universal set : A set which consists of all elements of different sets under consideration is called universal set.

Example : A ={ 1,2,3 }

B = {3,4,5,6,7,8}

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

In the above example U is the universal set.

Universal set is always denoted by U.

Link: https://youtu.be/8innwDI1bv8

6) Equal sets : Two sets are said to be equal if they have the same elements. All equal sets have same number of elements.

Example : A ={ month of the year having less than 30 days}

B={February}

Here sets A and B are equal sets.

7) Equivalent sets: Two sets are are said to be equivalent if the number ofelements in both the sets are equal.

Example : A= { 1,2,3}

B = { a,b,c}

In the above example set A and Set B are equivalent sets.

All equal sets are equivalent but equivalent sets may or may not be equal.

Link :https://www.youtube.com/playlist?list=PLmdFyQYShrjfi7EeDyHxr0jhoPXEOlFX0

8) Disjoint set : Two sets are said to be disjoint if they have no elements in common.

Example: A={1,3,5}

B = {2,4,6}

In the above examples Sets A and B have no elements are common. So sets A and B are disjoint sets

9) Overlapping sets : Two sets are said to be overlapping if they have one or more elements in common.

Example: A = {1,2,3,4}

. B= {3,4,5,6,7}

In the above examples sets A and B have common elements 3 and 4 .

So sets A and B are overlapping sets.

Cardinal number of of a set:

The number of distinct elements in a finite set is called cardinal number of the set. And is denoted by n (set name)

Example: A= { a,b,c,d,e}Cardinal Number =5

Denoted as n(A) = 5

Example : set of letters in the word ‘RAJASTHAN’B = { R,A,J,S,T,H,N}

n(B) = 7

Cardinal number of an infinite set is not defined.

Link: https://youtu.be/Voj2N0VVbmg

Test yourself:

• Q1 State whether the given set are finite or infinite.

• a) D ={ x|x is a multiple of 25}

• b) F ={ number of people in Surat}

• Q2 which of the following sets are empty?

• a) E ={score of a batsman who scored zero runs}

• b) R={x|x is a prime number less than 2}

Q3 write the cardinal number of the the following sets.

a) { letters in the word HIPPOPOTAMUS}

b) { 0 }

Check your answers

1 a) infinite set

b) finite set

2 a) not empty set

b) empty set

3 a) 9

b) 1

Worksheet to test yourself more

Answers :Q1. a) A ={2,4,6,8}

b) B ={ 3,6,9,12,15,18,21,24,27}

Q2. a) C ={ x|x is a prime number, x<100}

b) D ={x|x is a consonant in the word UNITED STATES}

Q3) a) A={ 2,3,5,7,11,13,17,19}

B ={ prime numbers less than 20}

b) A={ 0,1,2,3,4}

B={ whole numbers less than 5}

• c) A={ 1,5,7,35}

• B ={ factors of 35}

Q4. a) Infinite set

b)Finite set

c) Finite set

Q5. a) empty set ( because there is no number)

b) not empty set. ( because {1})

c) empty set ( because batsman have not done batting)

Q6. a) infinite set

b) infinite set

c) infinite set

Q7. a) 4

b) 4 (because numbers which are repeated are counted only once)

c) 10 (repeated elements are counted once)

Q8. a) false.

b) false

c) false

d) true

Q9. x = 9

Q10. a) Disjoint sets

A = { M,T, H,S} and B = { I } In sets A and B nothing is common so they are disjoint sets.

b) Overlapping sets

A = {0,1,2,3,…,14} and B ={ 1,2,3,4,6,12}. In sets A and B elements 1,2,3,4,6,12 are repeated. So they are overlapping .