#### 1.1 Introduction

- The theory of sets was developed by German mathematician Georg Cantor.

- Sets are used to define the concepts of relations and functions.

- The study of geometry, sequences, probability, etc. requires the knowledge of sets.

#### 1.2 Sets And Their Representations

- A set is a well-defined collection of objects.

- Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc.

- The elements of a set are represented by small letters a, b, c, x, y, z,

###### 1.2.1 Representation Of Sets

- There are two methods of representing a set:

- Roster or tabular form

- Set – builder form.

###### 1.2.1.1 Roster Form:

- In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces { }.

- Example: The set of all even positive integers less than 7 is described in roster form as { 2, 4, 6 }.

###### 1.2.1.2 Set-Builder Form

- In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.

- Example: In the set {a, e, i, o, u}, all the elements possess a common property, namely, each of them is a vowel in the English alphabet, and no other letter possess this property.

- Denoting this set by K, we write,

K = { *x : x *is a vowel in English alphabet }

##### Example:

- Write the set L = {1, 4, 9, 16, 25, . . . }in set-builder form.

##### Solution:

- We may write the set L as

L = {x : x is the square of a natural number}

- Alternatively, we can write

L = {x : x = n^{2 }, where n N}

#### 1.3 The Empty Set

- A set which does not contain any element is called the empty set or the null set or the void set.

- The empty set is denoted by the symbol or { }.

##### Example:

- D = {x : x
^{2 }= 4, x is odd}. Then D is the empty set because the equation x^{2 }= 4 is not satisfied by any odd value of x.

#### 1.4 Finite and Infinite Sets

###### Finite Sets

*A set which is empty or consists of a definite number of elements is called finite*.

##### Example:

- Let W be the set of the days of the week. Then W is finite.

###### Infinite Sets

*An infinite set is a set with an infinite number of elements.*

##### Example:

- The set of all integers, {….., -1, 0, 1, 2, …..}

#### 1.5 Equal Sets

- Two sets A and B are said to be equal

##### Example:

- Let A be the set of prime numbers less than 6 and P the set of prime factors of 30.

- Then A and P are equal, since 2, 3 and 5 are the only prime factors of 30 and also these are less than 6.

#### 1.6 Subsets

- A set P is said to be a subset of a set Q if every element of P is also an element of Q.

- It is expressed as P Q;The symbol stands for ‘is a subset of’.

##### Example:

- Let A = { a, e, i, o, u } and B = { a, b, c, d }. Is A a subset of B?

##### Solution:

- No, A is not a subset of B as every element of A is not an element of B.

#### 1.7 Power Set

- The collection of all subsets of a set A is called the power set of A.

- It is denoted by P(A).

##### Example:

- If A = { 1, 2 }, then P(A) = { , {1}, {2}, {1, 2} }

###### Note:

Cardinal Number :- The total no. of elements are known as Cardinal no.

- If A is a set with n(A) = m, then it can be shown that n[P(A)] = 2
^{m }.

#### 1.8 Universal Set

- A Universal Set is the set of all elements under consideration, denoted by capital U.

- All other sets are subsets of the universal set.

##### Example:

- Given that U = {5, 6, 7, 8, 9, 10, 11, 12}, list the elements of the following sets.

*A*= {*x*:*x*is a factor of 60}

*B*= {*x*:*x*is a prime number}

##### Solution:

- The elements of sets A and B can only be selected from the given universal set U.

*A*= {5, 6, 10, 12}

*B*= {5, 7, 11}

#### 1.9 Venn Diagrams

**A Venn diagram is a drawing in which circular areas represent groups of items which share a common property**.

- The drawing consists of two or more circles, each representing a specific group or set.

##### Example:

- Given U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = { 1, 2, 5, 6 } and B = { 3, 9 }, draw a Venn diagram to represent these sets.

##### Solution:

- Given

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

- A = { 1, 2, 5, 6 } and

- B = { 3, 9 }.

#### 1.10 Operations on Sets

- There are some operations which when performed on two sets give rise to another set. They are:

- Union of sets

- Intersection of sets

- Difference of sets

- Complement of a set

###### 1.10.1 Union of Sets:

- The union of two sets A and B is the set C which consists of all those elements which are either in A or in B (including those which are in both).

- The symbol ‘’ is used to denote the union.

- Symbolically, we write A B = {x : x A or x B}

- The union of two sets can be represented by a Venn diagram as shown in figure.

##### Example:

- Let A = { 2, 4, 6, 8 } and B = { 6, 8, 10, 12 }. Find A B?

##### Solution:

- We have A = { 2, 4, 6, 8 } and B = { 6, 8, 10, 12 }. Then A B = { 2, 4, 6, 8, 10, 12 }

##### Properties of the Operation of Union:

- Commutative law : A B = B A

- Associative law : (A B) C = A (B C)

- Law of identity element : A = A

- Idempotent law : A A = A

- Law of U : U A = U

###### 1.10.2 Intersection of Sets:

- The intersection of sets A and B is the set of all elements which are common to both A and B.

- The symbol is used to denote the intersection.

- Symbolically we write it as A B = {x : x A and x B}

- The shaded portion in the figure indicates the intersection of A and B.

##### Properties of Operation of Intersection:

- Commutative law: A B = B A

- Associative law: (A B) C = A (B C)

- Law of and U: =

- Idempotent law: A A = A

- Distributive law:

###### 1.10.3 difference of Sets:-

- The difference of the sets A and B in this order is the set of elements which belong to A but not to B.

- Symbolically, we write A – B and read as “A minus B”.

##### Example:

- Let A = { 1, 2, 3, 4, 5, 6 }, B = { 2, 4, 6, 8 }. Find A – B and B – A.

##### Solution:

- A – B = { 1, 3, 5 }, since the elements 1, 3, 5 belong to A but not to B

- B – A = { 8 }, since the element 8 belongs to B and not to A.

#### 1.11 Complement of a Set

- Let U be the universal set and A is a subset of U. Then the complement of A is the set of all elements of U which are not the elements of A.

- Symbolically, we write A’ to denote the complement of A with respect to U. Thus,

A’ = {x : x U and x A}

##### Example:

- Let U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } and A = { 1, 3, 5, 7, 9 }. Find A’.

##### Solution:

- 2, 4, 6, 8, 10 are the only elements of U which do not belong to A.

- Hence A’ = { 2, 4, 6, 8, 10 }

##### Properties of Complement Sets:

- Complement laws:

- A A’ = U

- A A’ =

- De Morgan’s law:

- (A B)’ = A’ B’

- (A B)’ = A’ B’

- Law of double complementation:

- (A’)’ = A

- Law of empty set and universal set

###### Formulae:

n(A B ) = n(A) + n(B) – n(A B)

n(A B C) = n(A) + n(B) + n(C) – n(A B) – n(B C) – n(A C) + (A B C)

##### Example:

- In a school there are 20 teachers who teach mathematics or physics. Of these, 12 teach mathematics and 4 teach both physics and mathematics. How many teach physics?

##### Solution:

- Let M denote the set of teachers who teach mathematics and P denote the set of teachers who teach physics.

n (M P) = 20,

n(M) = 12

n(M P) = 4

- To find n(P)

n (M P) = n(M) + n(P) – n(M P)

20 = 12 + n(P) – 4

n(P) =12

#### Thank You:- Prof Rajeev Tripathi

- Please email me at stephenhawking1982@gmail.com and help me to improve this

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