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Class – XI, CBSE MB1101: Sets

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What is a Set?

Class – XI MB1101: Sets

Important Definitions

Topic: Sets



What is a Set?

Class - XI MB1101: Sets


Topic: Sets

A set is a "well defined collection of objects, where the order in which the elements are listed is immaterial and while listing the elements repetition is not allowed.



What are the symbols for some important sets?



Topic: Sets



What are the symbols for some important sets?



Topic: Sets

N: the set of all natural numbersZ: the set of all integersQ: the set of all rational numbersR: the set of real numbersZ+ : the set of positive integersQ+ : the set of positive rational numbersR+ : the set of positive real numbers



What is a Notation?



Topic: Sets



What is a Notation?



Topic: Sets

If is an element of a set , we say that" belongs to " the Greek symbol(epsilon) is used to denote the phrase ' '. Thus, we write . .If ' ' is not an element of a set , we write

a Aa A

belongs to a bb A

b A

and read " does not belong to ".Thus, in the set = 1, 2, 3 , 1 but

4 .

bA

A A

A



How to represent a set in Roster or Tabular Form?



Topic: Representations of Sets



How to represent a set in Roster or Tabular Form?




In roster form, all the elements of a set arelisted, separated by commas and are enclosedwithin braces { }. For example, the set of alleven positive integers less than 7 is describedin roster form as {2, 4, 6}. The set of odd natural numbers is represented by {1, 3, 5, . . .}.The dots tell us that the list of odd numbers continue indefinitely.



How to represent a set in Set-Builder Form?






How to represent a set in 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. For example, in the set { , , , , }, all the elements possessa e i o u a common property, namely, each of them isa vowel in the English alphabet, and no other letter possess this property. Denoting this set by , we write

= { : is a vowel in English alphabet}.V

V x x



What is Empty set?



Topic: Empty Set



What is Empty set?



Topic: Empty Set

We define empty set also called null or void set, as a set which does not contain any element. We denote empty set by the symbol .



What are Finite and Infinite sets?



Topic: Finite and Infinite Sets



What are Finite and Infinite sets?



Topic: Finite and Infinite Sets

A set which is either empty or containsa definite number of element, is calleda finite set. A set which is not finite iscalled is an infinite set e.g. the set of all natural numbers is an infinite set

and

1, 2,3 is a finite set .A



What are equal sets?



Topic: Equal Sets



What are equal sets?



Topic: Equal Sets

Given two sets and , if every element of is also an element of and if every element of is also an element of , then the sets and are said to be equal. In this case we write = . If two

A BA B

B AA B

A B sets are equal then they have exactly the same elements. If two sets are not equal then they are called as unequal sets and we write A B



What are Subsets?



Topic: Subsets



What are Subsets?



Topic: Subsets

A set is said to be a subset of a set if every element of is also an element of .In other words, if whenever , then

. We use the symbol " " which means implies. Using this symbol, we

A BA B

A B a Aa B

can write the definition of as follows:

if .We read the above statement as " is a subset of if is an element of implies that is also an element of ". If is not a subset of

subsetA B a A a B

AB A A

B A

, we write .

BA B



What are Subsets?



Topic: Subsets

We use the symbol " " for two way implications, and read it as if and only also written as "iff"). So and It follows from the above definition that every set , i.e.,

A B A B B A

A is a subset of itself

. Since the empty set has no elements, we agree to say that is a subset of every set. We now consider some examples :Some of the obvious relations among the subsets of real numbers are

A A

N Z Q R



What is Super Set?



Topic: Subsets



What is Super Set?



Topic: Subsets

Let and be two sets. If and , then is called a proper subset

of and is called a superset of A. For example, = {1, 2, 3} is a proper subset of = {1, 2, 3, 4}.

A B A BA B A

B BA

B



What is Singleton Set?



Topic: Subsets



What is Singleton Set?



Topic: Subsets

If a set has only one element, we call it a . Thus,{ a } is a singleton set.

Asingleton set



Explain Intervals as a subset of R?



Topic: Subsets



Describe Intervals as a subset of R?



Topic: Subsets


Let , R and . Then the set of real numbers

: is called an open interval and is

denoted by ( , ). All the points between and belong to the open interval ( , ) but , themselves do n

a b a by a y b

a b a ba b a b

ot belong to this interval. The interval which contains the end points also is called and is denoted by [ , ]. Thus, , : .

We can also have intervals closed at one end and open

closed intervala b a b x a x b

at the other, i.e., , : is an

from to , including but excluding ., : is an open interval from to

including but excluding .

a b x a x b open

interval a b a ba b x a x b a b

b a





Topic: Subsets


These notations provide an alternative way of designating the subsets of set of real numbers. For example, if 3, 5 and 7, 9 ,

then . The set 0, defines the set of

non-negative real numbers,

A B

A B

while set ,0

defines the set of negative real numbers. The set , describes the set of real numbers as a line

extending from to . The number ( ) is called the length of any of the intervals

b a

( , ),

[ , ], [ , ) or ( , ].a b

a b a b a b





Topic: Subsets


On real number line, various types of intervals described above as subsets of , are shown below:Graphically, an open interval 3,1 is shown as:

R

The interval which contains the end points also is called closed interval and is denoted by ,

Thus, , : . Graphically, a closed

interval 3,1 is shown as:

a b

a b x a x b





Topic: Subsets


We can also have closed at one end and open at the other, i.e., , : is an half-open

interval from to , including but excluding ., : is an half-open interval f

half - open intervalsa b x a x b

a b a ba b x a x b

rom to

including but excluding . So, the half open interval 2,4

is shown graphically as:

a b

b a


What is Power Set?



Topic: Power Set



What is Power Set?



Topic: Power Set

The collection of all subsets of a set is called the of . It is denoted by

( ). In ( ), every element is a set.Thus, as in above, if A = { 1, 2 }, then

( ) = { , { 1 }, { 2 }, { 1,2 }}

Apower set A

P A P A

P A

2

.Also, note that the number of elements

in ( ) = [ ( )] = 4 = 2 .In general, if is a set with ( ) = , then

it can be shown that [ ( )] = 2 .m

P A n P AA n A m

n P A



What is Universal Set?



Topic: Universal Set



What is Universal Set?



Topic: Universal Set

A universal set is the collection of all objects in a particular context. All other sets in that framework constitute subsets of the universal set, which is denoted as an uppercase italic letter . Th

Ue objects themselves are known as

elements or members of . U



What are Venn Diagrams?



Topic: Venn Diagrams








Most of the relationships between sets can be represented by means of diagrams which are known as .In Venn diagrams, the elements of the sets are written in their respective circles. The

Venn diagrams

Venn diagram with three overlapping circle makes 7 different regions as shown below:







Let us try to understand what these regions indicate:Region 1 contains elements that are in but not in or .

Thus 1, if , and .Region 2 contains elements that are in but not in

A B C

x x A x B x CB A

or .

Thus 2, if , and .Region 3 contains elements that are and but not in .

Thus 3, if , and .Region 4 contains elements that are in but not in or .

Thus 4, i

C

x x B x A x CA B C

x x A x B x CC A B

x

f , and .Region 5 contains elements that are and but not in .

Thus 5, if , and .Region 6 contains elements that are and but not in .

Thus 6, if , and .R

x C x A x BA C B

x x A x C x BB C A

x x B x C x A

egion 7 contains elements that are in , and .

Thus 7, if , and .

A B C

x x A x B x C


What is Union of Sets?



Topic: Operations on Sets








Let and be any two sets. The union of and is the set which consists of all the elements of and all the elements of , the common elements

being taken only once. The symbol ' ' is used to denot

A B ABA B

e the . Symbolically, we write

and usually read as ' union '.Thus, we can define the union of two sets and as the set C which consists of all those elements which are either in A or in

union A BA B

A B

B (including those which are in both). In symbols, we write

: or .A B x x A x B







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

The shaded portion in Figure represents .A B


What is Intersection of Sets?











The intersection of sets and is the set of all elements which are common to both and . The symbol ' ' is used to denote the .

The intersection of two sets and is the set of all t

A BA Bintersection

A B

hose elements which belong to both and .

Symbolically, we write : and .

Thus, we can define the intersection of two sets and as the set of all those elements which belong both A and B.

A BA B x x A x B

AB







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


What is Difference of Sets?











The difference of the sets and in this order is the set of elements which belong to but not to . Symbolically, we write and read as" minus ".Using the set builder notation, we can rewrit

A BA B

A B A B

e the definition

of difference as : and .

The difference of two sets and can be represented by Venn diagram as shown in the figure.

A B x x A x B

A B

The shaded portion represents the difference of the two sets and .A B







The sets , and are mutually disjoint sets, i.e., the intersection of any of these two sets is the null set as shown in the figure

A B A B B A


What is Complement of a Sets?






What is Complement of a Sets?





Let be the universal set and is a subset of . Then the complement of is the set of all elements of which are not the elements of A. Symbolically, we write ' to denote the complement of wit

U AU A

UA

A

h respect

to . Thus, : and . Obviously . So the complement of a set can also be seen as difference between a universal set and the set A.

U A x x U x AA U A

A


Fact # 1

Two sets A and B are said to be equal if they have exactly the same elements.


Important Facts

Topic: Equal Sets



Fact # 2

A set A is said to be subset of a set B, if every element of A is also an element of B.


Important Facts

Topic: Subsets



Fact # 3

Intervals are subsets of R.


Important Facts

Topic: Subsets



Fact # 4

A power set of a set A is collection of all subsets of A. It is denoted by P(A).


Important Facts

Topic: Power Sets



Fact # 5

The union of two sets A and B is the set of all those elements which are either in A or in B.


Important Facts




Fact # 6

The intersection of two sets A and B is the set of all elements which are common. The difference of two sets A and B in this order is the set of elements which belong to A but not to B.


Important Facts




Fact # 7

The complement of a subset A of universal set U is the set of all elements of U which are not the elements of A.


Important Facts

Topic: Complement of a Set




Important Formulae


1. Complement law:

( ) ( ) i A A Uii A A




Important Formulae


2. De Morgan's law:

( )

( )

i A B A B

ii A B A B




Important Formulae


3. Law of double complementation:

A A




Important Formulae


4. Laws of empty set and universal set

and .U U




Important Formulae


5. Commutative law

, A B B A A B B A




Important Formulae


6. Law of identity element

, , A A A U A A




Important Formulae


7. Idempotent law

, A A A A A A




Important Formulae


8.

U A U




Important Formulae


9. Associative Laws:

, A B C A B C A B C A B




Important Formulae


10. Distributive Laws:

, A B C A B A C A B C A B A C




Important Formulae


11. Let and be finite sets. If thenA B A B

n A B n A n B




Important Formulae


12. In general, if and are finite sets, thenA B

.n A B n A n B n A B




Important Formulae


13. For any three finite sets , , A B C

.

n A B C n A n B n C n A B n B C

n A C n A B C



How to change the representation of the set from Set-Builder to Roster Form?

Class – XI,CBSE

Important Procedures



MB1101: Sets



Class – XI,CBSE




: Understand the description of the elements of the set. Step1

MB1101: Sets



Class – XI,CBSE





MB1101: Sets

: Identify each member of the set one at a time. Step2



Class – XI,CBSE





MB1101: Sets

: Identify each member of the set one at a time. Step2: If you can see a pattern after finding a few members, use an ellipsis to write the set in

Roster form so that everyone can see the pattern and guess ALL elements.

Step3



Class – XI,CBSE





MB1101: Sets

: Identify each member of the set one at a time. Step2: If you can see a pattern after finding a few members, use an ellipsis to write the set in

Roster form so that everyone can see the pattern and guess ALL elements.

Step3

: If there are just a few elements (usually less than 10) then you need to mention all

elements in the Roster form.

Step4


How to change the representation of the set from Roster to Set-Builder Form?

Class – XI,CBSE




MB1101: Sets



Class – XI,CBSE




MB1101: Sets

: Find a pattern among all members of the set.Step1



Class – XI,CBSE




MB1101: Sets

: Find a pattern among all members of the set.Step1: Find out what are the common characteristics of the members.Step2



Class – XI,CBSE




MB1101: Sets

: Find a pattern among all members of the set.Step1: Find out what are the common characteristics of the members.Step2: Use a group of statements to describe these common characteristics in set-builder form.

This conversion is not always easy because the description of the elements of the set may not be unique.

Step3


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