We shall discuss introduction to sets,What is Sets , kinds of sets, representation of sets, subset , power set , universal set ,cardinal number , union, intersection of sets , disjoint sets , complement and difference of sets .And we shall be able to solve Puzzles based on set like given below

## WHAT IS SETS

Def : Any collection of well defined and distinct objects is called a set.

By well defined set/object we mean , for a given set and object, it must be possible to decide whether the object belonging to the set or not . i .e it is accepted by everyone as an element for that Set or not .The object in a set are called elements of a set or Members.

Set are usually denoted by Capital letter A , B , C , D Etc, and their element are represented by small letters.

For any set A let 'a' , 'b' and 'c' are members of that Set, then a ∈ A , b ∈ A and c ∈ A . This special character ∈ is read as belongs to symbol . So a ∈ A will be read as " a belongs to A" , b ∈ A means "b belongs to A" and c ∈ A read as " c belongs to A".

#### Here are some examples of sets given below

(1) The collection of vowels in English Alphabet.

This is Set because every one will accept the letters from a , e , i , o , u as the Vowels in English Alphabet

This is Set because every one will accept the letters from a , e , i , o , u as the Vowels in English Alphabet

(2) The collection of Mountains in India.

(3) The collection of all districts in India.

(4) The collection of even numbers .

(5) The collection of all the M L As and M Ps in India .

(6) The collection of all the football players of India.

(7) The collection of hundred novels of Indian writers.

(8) The collection of Natural numbers from 100 to 55555.

(9) The collection of prime numbers less than 100.

(10) The collection of Prime Ministers of India .

All those person who have accepted the post of Prime Minister are the elements of the set of Prime Ministers of India .On this point there will be no debate,The debate can be on the topic that particular Prime Minister is/was a good , popular or is/was a caring Prime minister.

All those person who have accepted the post of Prime Minister are the elements of the set of Prime Ministers of India .On this point there will be no debate,The debate can be on the topic that particular Prime Minister is/was a good , popular or is/was a caring Prime minister.

These were examples of set as all the examples given above were well defined and easily acceptable to everyone.

#### Here are some examples which are not Set

(1) The collection of beautiful Mountains in India.

Because the word beautiful is not acceptable to all , Some peoples can says that these Mountains are beautiful But at the same time some other peoples can says that particular Mountains are not beautiful in their opinions .

(2) The collection of

Similar is the case with Popular Leaders, as some people likes MLAs Or MPs But at the same time many other peoples hate them. So the word Popular is not acceptable to all .

**Popular**M L A and M P in India .Similar is the case with Popular Leaders, as some people likes MLAs Or MPs But at the same time many other peoples hate them. So the word Popular is not acceptable to all .

(3) The collection of

**best**football players of India.
(4) The collection of an

**Interesting**novels of Indian writers.
(5) The collection of

**handsome**students in any class .
(6) The collection of

**dangerous**animals in this world.
(7) The collection of

**rich**person in India.
(8) The collection of an

**easy**subjects.
(9) The collection of an i

**ntelligent**students in any class.
(10) The collection of

**beautiful**animals .
These were not examples of set as all the examples given above were not well defined and were not easily acceptable to everyone. Because the some mountains are beautiful for some person but at the same time ,same mountains are not beautiful for some persons , Similarly some peoples find any book interesting but at the same time same book may not be interesting for others.

##

Representation of Sets

The set can be represented in two types

###
1 Roster Form or Tabulation Form

2 Set builder form or Rule form

### Roster Form

To represent elements in Roster form ,we separate every element by comma and all the elements are listed in this form.

For example

Let A is the collection of even Natural Numbers . Then

A = {2, 4, 6, 8, 10, ..............}

Let P is the collection of prime numbers less than 100

P = {2, 3, 5 , 7, 11, 13, 17, 23, 29, ........, 97}

Let V is the collection vowels in English Alphabet.Then

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

Let S is set of square of the natural number between 1 to 10 Then

S= {1, 4, 9, 16, 25, 36, 49, 64, 81, 100}

### Set Builder form

In this form of set , we write the elements of set by special rule, and that rule must be satisfied by every element of the set in this form all the elements of the set are not written. Given below are Set builder notation examples.

Let A is the collection of even Natural numbers . then

A = {x : x = 2n , n ∈ N }

Let V is the collection of vowels in English Alphabet.Then

V = { x : x is a vowel in English Alphabet }

Let S is set of square of the natural number between 1 to 10 Then

S= { x : x =

*n*^{2}: n ∈ N and 1 ≤ n ≤ 10 }### Empty (Null Set or Void Set)

When a set having No element in it then that Set is Called an

Empty Set and Empty or null set symbol is denoted by Ñ„ or { }.

Empty set examples are given below

(1) Ñ„ = {x : x ∈ N , 5 < x < 6}

(2) Ñ„ = {x : x ∈ R ,

*x*^{2}+3 = 0 }
(3) Ñ„ = { x ; 99 > x and x > 101}

### Non Empty Set

Any set which contain at least one element is called Non empty set .

Some examples of Non Empty set are given below

(1) A = { 0 }

(2) B = { x : x ∈ N , 5 < x < 8}

(3) C = { x ; 99 < x and x < 105}

(4) D = { x : x ∈ R ,

*x*^{2}- 3 = 0 }##
**Singleton Set**

Any set which contain only one element is called

**Singleton Set .****e. g**

(1) A = { 0 }

(2) B = { x : x ∈ N , 6< x < 8}

(3) C = { x ; x ∈ R , 99 < x and x < 101}

(4) D = { x : x ∈ R , x - 3 = 0 }##
**Finite Set **

The set in which total numbers of elements can be counted, is called Finite set.

Some of the Finite set examples are given below

(1) The set of all the countries in the world.

(2) The set containing all the prime number less than 1000.

(3) C = { x ; x ∈ N , 99 < x and x < 109}.

(4) D = { x : x ∈ R ,

*x*^{2}- 3 = 0 }.
(5) E = The set of all the even Natural Numbers less than 100.

##
**Infinite Set**

The set in which total numbers of elements can

do not come to an end ,is called Finite set.

**Not**be counted, i. e. their counting of elementsdo not come to an end ,is called Finite set.

Here are Infinite set examples given below

(1) The set of all points in a plane.

(2) The set containing all the prime number .

(3) C = { x ; x ∈ R , 99 < x and x < 109}.

(4) D = Set of lines parallel to given line

(5) E = The set of all the even numbers greater than 100.

(6) S = Set of stars in the sky,

(6) S = Set of stars in the sky,

as the stars can nit be counted , they are numberless.

## Cardinal Number of a Set

The numbers of distinct elements present in any finite set is called cardinal number of finite set. And it is denoted by n(A) .

If A = { 1 , 3 , 5 , 8 , 9 } then n(A) = 5 and

if

B= {a , b , c , d , e , f , g , h , i , j } then n(B) = 10 and if

C = {1 , 2 , 3 , 4 , 5 , 6 , ...... , 100 }then n(C) = 100

Because The set A , B and C have 5 ,10 and 100 elements respectively .

## What is a subset

The set B is said to be the subset of a set A if every element of set B is also an element of set A . Subset symbol is denoted by ⊆ , if we write A ⊆ B that means A is subset of B and if we write B ⊆ A means B is subset of A .

If A = { 1 , 3 , 5 , 8 , 9 } and B = { 1 , 3 , 5 } then B is called subset of A as all the elements of set B are in Set A.

##

Subset and proper subset

Any set B is called proper subset of set A if every element of set B is an element of set A whereas every element of set A is

**not**an element of Set B . It is denoted by ⊂ .
Let A = { 1 , 2 , 3} and B = { 1 , 2 } and C = { 1 , 3 } then B ⊂ A and also C ⊂ A . It is to be noted that A ⊆ A i.e. every set is a subset of itself but it is not a proper subset. As we know that ⊂ is notation for Proper subset symbol .

## Power Set

If we form a set which consist of all the subset of a given set A , then that set is called Power Set and is denoted by P(A) .

Let us consider Power set example if A = { a , b , c } then

P(A) = { Ñ„ , { a } , { b } , {c} , {a , b}, { a , c} , { b , c } , { a , b , c } }

## Equal Sets

Two sets are said to be equal to each others if they have same and same numbers of elements ,

Given below is an example of equal sets examples

If A = { 1 , 2} and B = {x : x ∈ N , 1 ≤ x ≤ 2 } , Then these sets A and B are equal to each other because both the sets have same numbers of elements and also same elements 1 and 2 .

## Equivalent sets

Two sets are said to be equivalent if both the sets have same numbers of elements , And Equivalent Sets need not be equal to each other but have same cardinal numbers .

e.g if A = { a , b , c } and B = { 1 , 2 , 3 } are Equivalent Sets

**Comparable Sets**

Two sets are said to be comparable if one of them is a subset of other i.e. either A ⊆ B or B ⊆ A .

A = { a , b , c } and B = { b , c } are comparable sets.

**Universal Set**

A set that contains all the sets under consideration , and is denoted by U .

When we are using set containing real numbers , then Set R is called Universal set .

If we are considering set of equilateral triangles , then Set of Triangles is called Universal set .

Let us understand it with Universal set example

If A = { 2 , 4 , 6 }, B = { 1 , 2 , 3 , 4 } and C { 5 , 6 , 8 , 9 , 10 }

then Set U = { 1 , 2 , 3 , 4 , 5 , 6 , 8 , 9 ,10 } is called Universal set.

## Sets and venn diagrams

Let us discuss below algebra of sets with the help of some examples.

## What is union set

The union of two sets A and B is the set which contains all those elements which are either in set A or in from set B or in both A and B. It is denoted by A U B .

Thus A U B = { x : x ∈ A or x ∈ B }If A = { 1 , 2 , 5 , 6 } and B = { 8 , 9 , 10 } then

A U B = {1 , 2 , 5 , 6 , 8 , 9 , 10 } , as this set contains the elements either from Set A or from Set B.

##

Again if A = { a , b , c , d, e , f , g } and B = { a , b , c , d }
A U B = { a , b , c , d, e , f , g } = A
Thus the union of a given set and its subset is always the given set

Again if A = { a , b , c , d, e , f , g } and B = { a , b , c , d }

##
**Intersection of sets**

Intersection of two sets A and B is the set which contains all those elements which are in both the set A and Set B . It is denoted by A ∩ B . So ∩ is called intersection symbol .

Thus A ∩ B = { x : x ∈ A and x ∈ B }

If A = { 1 , 2 , 5 , 6 } and B = { 5 , 6 , 10 }

then A∩ B = { 5 , 6 }

Again if A = { a , b , c , d, e , f , g } and B = { a , b , c , d }

then A∩ B = { a , b , c , d } = B

Thus the intersection of a given set and its subset is always the subset of a given set .

So these were some of examples of union and intersection examples.

##
**Disjoint Sets**

Two sets are said to be disjoint sets if they have no common element. i.e. if their intersection is a null set A⋂ B = { } = Ñ„

if A = { a , b , c , d, e , f , g } and B = { h , i , j , k , l , m } then these sets A and B are disjoint as they have no common element

i. e A∩ B = Ñ„ .

##
**Difference of Sets**

The difference of two sets A and B is the set of all those elements of A which which are

**not**in B . It is denoted by A - B.

Thus A - B = {x : x ∈ A but x ∉ B }

If A = { 2 , 3 , 4 , 5 , 6 } and B = { 5 , 6 , 7 , 8 } then

A - B = { 2 , 3 , 4 }

Similarly The difference of two sets B and A is the set of all those elements of B which which are

**not**in A. It is denoted by B - A.

Thus A - B = {x : x ∈ B but x ∉ A }

If A = { 2 , 3 , 4 , 5 , 6 } and B = { 5 , 6 , 7 , 8 } then

B - A = { 7 , 8 }

##
**what is complement of a set**

If we have a universal set U and a given set A ⊂ U then complement of a set A is the set which contains all those elements of universal set U which are in not in A . It is denoted by

*A*^{c}
Thus

*A*^{c }= { x : x ∉ A and x ∈ U }## Some Important Results

*U*

^{c }= Ñ„

*Ñ„*

^{c }=

*U*

*U*

^{c }= Ñ„

*(*

*)**A*^{c}

^{c = }

^{ }

*A*

*A*U

*A*

^{c = }

*U*

*Ñ„*

*A**∩ A*^{c = }*(A*U

*B )*

^{c = }

*A*

^{c}

*∩*

*B*

^{c De Morgan's Law}

^{}

^{(A ∩ B ) c = A c U B c }De Morgan's Law

## Practical Application of Sets

**If A and B are disjoint sets**

n

*(A*U*B ) = n(*A) + n(B)## If A and B are not disjoint sets

n (A U B ) = n(A) + n(B) - n (A ∩ B )## Question

In a class of 50 students , 35 opted Mathematics 25 opted Biology then how many opted both Mathematics and Biology ?

## Solution

##

Here n ( M U B ) represents the total ( sum ) of students in a class and n ( M ∩ B ) represents the common students who have opted both the subjects Biology and Mathematics in that class.

Here we can use this formula

n ( M U B ) = n ( M ) + n ( B ) - n ( M ∩ B ) --- -- -- (1)

where n (M U B ) = 50 , n ( M ) = 35 , n( B ) = 25 ,

n ( M ∩ B ) = ?

so putting the above values in (1)

50 = 35 + 25 - n ( M ∩ B )n ( M ∩ B ) = ( 35 + 25 ) - 50

## n ( M ∩ B ) = 60- 50

n ( M ∩ B ) = 10

## Question

## Solution

Here n ( E U H ) represents the total ( sum ) of people in a village and n ( E ∩ H ) represents the common Reader who read both the News Papers Hindi and English in that village .

n ( H U E ) = n ( H ) + n ( E ) - n ( H ∩ E ) --- -- -- (1)

where n (H ∩ B ) = 150 , n ( H ) = 350 , n( E ) = 235 ,

n ( H U B ) = ?

Here we can use this formula

n ( H U E ) = n ( H ) + n ( E ) - n ( H ∩ E ) --- -- -- (1)

where n (H ∩ B ) = 150 , n ( H ) = 350 , n( E ) = 235 ,

n ( H U B ) = ?

so putting the above values in (1)

n ( H U E ) = 350 + 235 - 150n ( M U B ) = ( 350 + 235 ) - 150

## n ( M U B ) = 585 - 150

n ( M U B ) = 435

So there are 435 Peoples who read News Papers .

##

##

##

##

So there are 435 Peoples who read News Papers .

##
**Question **

##
In a group of 50 teachers , 30 teachers drinks coffee and 25 drinks both coffee and tea, How many teachers drinks tea and how many drinks tea only

##
**Answer**

Here C and T represents Coffee and Tea Respectively . Therefore Total numbers of teachers in that group are 50 i.e. n (T U C ) , and numbers of teachers who take both drinks are 30 . i . e.

n ( C ∩ T ) = 25 And n( C ) = 30 so n (T) = ? and Teachers drinks tea only will be represented with n ( T - C ) ,which we have to find

n ( T U C ) = n ( T ) + n ( C ) - n ( T ∩ C ) --- -- -- (1)

50 = n (T) + 30 - 25

n ( T ) = 45

It means that 45 Teachers drinks tea , this implies that in these 45

Teachers there are some Teachers who drinks both coffee as well . so by the statement how many drinks tea only , we have to drop those Teachers who drinks both ,

n( T - C ) = n ( T ) - n ( T ∩ C )

## n( T - C ) = 45 - 25

## n( T - C ) = 20

##
**Conclusion**

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