algebra b c a a ∩ b b ∩ c a ∩ c a ∩ b ∩ c sets jadhav s.s. m.s.v.satara standard ix...

Algebra

B

CA

Catego

ry 1

Catego

ry 2

Catego

ry 3

Catego

ry 4

0

2

4

6

Series 1Series 2Series 3

A ∩ B B ∩ C

A ∩ C

A ∩ B∩ C

Sets jadhav s.s. M.S.V.Satara

STANDARD IX

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Std-9th

Sub-Mathematics

Chapt.-Sets

Brilliant students

in my class(b)

My numbers2,10,6,11,4,8

(d)

Happy people in my town

(c)

My objects(e)

Richest personTown

(f)

My bouquet

(a)



Std-9th

Sub-Mathematics

Chapt.-Sets

• The objects in the posters of (a),(d),(e)are clearly seen.• Such collections are well defined collections.

• The names of students or persons in posters (b),(c), (f) are not one and the same , such collections are not well defined collections.

• A well defined collection of objects is called a set.• Individual object in the set is called an element or member of

the set.• Sets are denoted by capital alphabets e.g. A,B,C, etc.The

elements of sets are generally denoted by small alphabets e.g. a,b,c etc.

• If x is an element of the set X then we write it as xє X and if x is not an element of set X then we write x є X.

• (є:belongs to , є:does not belongs to)

•



Std-9th

Sub-Mathematics

Chapt.-Sets

(a) N= the set of nonnegative integers or natural number={1,2,3,...}

(b) W= the set of whole numbers= {0,1,2,3,...}

(c) I=the set of integers={…,-3,-2,-1,0,1,2,3,...} (d) Q=the set of rational numbers

(e) Q+=the set of positive rational numbers

(f) R=the set of real numbers

Common notations



Std-9th

Sub-Mathematics

Chapt.-Sets

Sets

Methods of writing sets•(a) Listing method(Roster form)•(b)Rule method(Set builder form)



Std-9th

Sub-Mathematics

Chapt.-Sets

Sets• LISTING METHOD(Roster form)• In this method ,first write the name of the set ,

put is equal sign and write all its elements enclosed within curly brackets{ }.

• Elements are separated by commas.• An element , even if repeated , is listed only once.• The order of the elements in a set is immaterial.



Std-9th

Sub-Mathematics

Chapt.-Sets

• e .g. 1) The set of all natural numbers less than ten.

• A={1,2,3,4,5,6,7,8,9}• 2) The set of colours in the rainbow.• B={red ,orange ,yellow ,green,blue,indigo,voilet} • 3) C =The set of letters in the word

‘MATHEMATICS’• C={m , a , t ,h ,i ,c , s}



Std-9th

Sub-Mathematics

Chapt.-Sets

Rule Method(Set builder form) In the set builder form we describe the elements of

the set by specifying the property or rule that uniquely determines the elements of the set.

Consider the set P={1,4,9,16,}P={x x =n2,nєN , n=1,2,3,4,5}

In this notation ,the curly bracket stands for ‘the set of ’,

vertical line stands for, such that.Here ‘x’ represents each elements of that set.

And read as “P is the set of all x such that x is equal to n2,where n є N and n is less than or equal to 5.”



Std-9th

Sub-Mathematics

Chapt.-Sets

• e . g. 1)The set of prime numbers from 1 to 20• A={2,3,5,7,11,13,17,19}• This can be written in set builder form as :• A={x x is a prime number less than 20 }• 2)B={-7,7}• B={x:xis a square root of 49}



Std-9th

Sub-Mathematics

Chapt.-Sets

Venn-Diagrams :Many ideas or concepts are better understood with help of diagrams . Such presentation used for sets is called Venn-diagram .For this use the

closed figure and elements of the sets represented by points in that closed figure.

A={ 1,2,3,4,5} can be represented as.1

.2.3

.4

.5.1 .2 .3

.4

.5.1

.2 .3.4.5

.1.2

.3.4

.5or

or

or

A AA

A



Std-9th

Sub-Mathematics

Chapt.-Sets

Sets• Types of sets:

• 1) Empty set or Null set : • A set which does not contain any element

is called Empty or Null set. It is denoted by {} or Φ• e.g. The set of men whose heights are more than

5meter.

• 2) Singleton Set :• A set containing exactly one element is

called a Singleton set.• e.g. 1)P={x:x is a natural number,4<=x<=6} • 2)E={0}



Std-9th

Sub-Mathematics

Chapt.-Sets

Sets• Finite set : • If the counting process of elements of a set

terminates , such a set is called a finite set.• e.g. B={1,2,3,…,200000}• D={a,e,i,o,u}

• Infinite set :• If the counting process of elements of a set

do not terminates at any stage , such a set is called a Infinite set.

• e.g. N={1,2,3,4,…}



Std-9th

Sub-Mathematics

Chapt.-Sets

• Subset :

.6.3 .7

.1.2

.5

.4A

B Consider

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

B={3,6,7}

Here ,every element of set B is an element of the set AIf every element of set B is an element of set A then set B is said to be the Subset of set A and we write as B A.



Std-9th

Sub-Mathematics

Chapt.-Sets



Std-9th

Sub-Mathematics

Chapt.-Sets

SetsIf B is a subset of A and the set A contains at

least one element which is not in the set B, then the set B is the Proper subset of set A.

It is denoted as B U A. In this case the set A is said to be the Super set

of set of the set B and is denoted as B U A.Note:1) Every set is a subset of itself. 2)Every set is a subset of every set.

Sets

Universal set : A suitable chosen non-empty set of which all

the sets under consideration are the subsets of that set is called the Universal set.

e.g. If A={2,3},B={1,4,5},C={2,4} then U={1,2,3,4,5} can be taken as the universal set of the sets A,B and C.



Std-9th

Sub-Mathematics

Chapt.-Sets

Sets

(a)Equality: If A is subset of B and B is subset of A,

then A and B are said to be equal sets and are denoted by A=B.

e.g. If A={2,4,6,8},B={4,8,2,6,} then A=B.

Operations on sets



Std-9th

Sub-Mathematics

Chapt.-Sets

Operations on sets :(b)Intersection of sets: If A={1,2,3,4, 6,7} and B={2,4,5,6,8} then C={2,4,6} is called the intersection of the

sets A and B.

The set of all common elements of A and B is called the intersection A and B.

.1

.7

.3 .4.2

.6 .8

.5A BA U B



Std-9th

Sub-Mathematics

Chapt.-Sets

Operations on sets :Disjoint sets: Let A={1,2,3,4} and B={5,6,7,8} Here both sets A and B have no common

elements . Therefore set A and B are Disjoint sets. A ∩ B={ } or Φ

.1

.2.3 .4

A B.5 .6 .7

.8



Std-9th

Sub-Mathematics

Chapt.-Sets

Properties of Intersection of sets:1)A ∩ B =B∩ A (commutative property) 2)A ∩ (B ∩ C) =(A ∩ B) ∩ C (associative property)3) A ∩ B ⊆ A; A ∩ B ⊆ B 4)If A ⊆ P; B ⊆ P then A ∩ B ⊆ P5)If A ⊆ B then A ∩ B=A. If B ⊆ A then A ∩ B = B6)A ∩ Φ = Φ and A ∩ A =A



Std-9th

Sub-Mathematics

Chapt.-Sets

Operations on sets: (c) Union of sets: Let A={1,2,3,4} and B={4,5,6,1,8} be the sets. If we write set C , which contains all the elements

of A and B together is called the Union of sets A and B.As follows

C={1,2,3,4,5,6,8}

.4.1.2

.3.5 .6

.8A U B

A B



Std-9th

Sub-Mathematics

Chapt.-Sets

Properties of Union of sets:

1)A U B=B U A2)A U (B U C)=(A U B) U C3)A ⊆ (A U B) and B ⊆ ( A U B)4)If A ⊆ B then (A U B) =B and (B U A) =A5)(A U ø ) =A 6)(A U A)=A



Std-9th

Sub-Mathematics

Chapt.-Sets

Operations on sets :(d)Difference of two sets : Consider the following two sets. A={1,2,3,4,5}and B={1,2,6,7,8} If we write the set C , which contains

all the elements in set A but not in set B is called the Difference of sets A and B .As

C={3,4,5}

.1

.2.3

.4

.5 .6.7.8

A B

A-B



Std-9th

Sub-Mathematics

Chapt.-Sets

Properties of Difference of sets:

1)A - B ≠ B - A2)A-B ⊆ A 3)If A ⊆ B, then A –B= ø 4)If A ∩ B= ø, then A - B =A



Std-9th

Sub-Mathematics

Chapt.-Sets

Operations on sets :

(e)Complement of set :

Consider U={ x:x is a natural number , x<9} A={2,3,5} First we U in the roster form U={1,2,3,4,5,6,7,8} then

U-A ={1,4,6,7,8} Now if we observe (U-A).

It contains all those elements of U which are not in A.

Here, (U-A) is called the complement of A . It is denoted by A, or Ac .

UA.2 .3

.5

.1

.4

.6

.7.8(U-A) or Ac



Std-9th

Sub-Mathematics

Chapt.-Sets

algebra b c a a ∩ b b ∩ c a ∩ c a ∩ b ∩ c sets jadhav s.s. m.s.v.satara standard ix...

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