PRESENTED BY AKASH & VICKY CLASS – 11TH - SCIENCE D.A.V. PUBLIC SCHOOL SECL KORBA CHHATTISGARH

The tongue has no bones,but is strong enough

To break a heart.so be careful with your words

(1)- INTRODUCTION OF SETS

(2)HISTORY OF SETS

(3)TYPES OF SETS(4)SETS FUNCTIONS

(5)VENN DIAGRAM(6)EXAMPLES OF

SETS(7)ACKNOWLEDGEM

ENT

*IN THESE DAYS OF CONFLICT BETWEEN ANCIENT AND MODERN STUDIES; THERE MUST SURELY BE SOMETHING TO BE SAID FOR A STUDY WHICH DID NOT BEGIN WITH PYTHAGORAS AND WILL NOT END WITH EINSTEIN: BUT IS THE OLDEST AND THE YOUNGEST. – G.H. HARDY*

SETS ARE USED TO DEFINE THE CONCEPTS OF RELATION AND FUNCTIONS.THE THEORY OF SETS WAS

DEVELOPED BY GERMAN MATHEMATICAN GEORGE CANTOR(1845-1918). HE FIRST ENCOUNTERED SETS WHILE WORKING ON “PROBLEMS ON TRIGNOMETRIC SERIES”. IN THIS CHAPTER. WE DISCUSS SOME BASIC

DEFINITIONS & OPERATIONS INVOLVING SETS.

Empty Sets (Null Sets) Finite And Infinite Sets Equal Sets Singleton Sets Equivalent Sets Power Sets Universal Sets

SETS

A = {1, 3, 2, 5}

n(A) = | A | = 4

Sets use “curly” brackets

The number of elements in Set A is 4

Sets are denoted by Capital letters

A3A7

3 is an element of A

7 is not an element of A

A set is a distinct collection of objects. The objects are called elements.

{1, 2, 3, 4} = {2, 3, 1, 4}Order does not matter. If a set contains the same elements as another set, the sets are equal.

{1, 3, 2, 3, 5, 2} We never repeat elements in a set.{1, 3, 2, 5}

This symbol means "is a subset of"

This is read "A is a subset of B". A BA = {1, 2, 3} B = {1, 2, 3, 4, 5}

{1, 2, 3, 5}

In ascending order

If a set doesn't contain any elements it is called the empty set or the null set. It is denoted by or { }. NOT {} It is agreed that the empty set is a subset of all other sets so: where is any set.A A

List all of the subsets of {1, 2, 3}.

Notice the empty set is NOT in set brackets.

{1} {2} {3} {1, 2} {1, 3} {2, 3} {1, 2, 3}

A

?

Number of Elements in Set

Possible Subsets Total Number of Possible Subsets

1. {A} {A} 2

2. {A , B} {A , B} {A} {B} 4

3. {A , B , C} {A , B , C} {A , B} {A , C} {B , C} {A} {B} {C}

8

4. {A , B , C, D} {A , B , C , D} {A , B , C} {A , B , D} {A , C , D} {B , C , D} {A , B} {A , C} {A , D} {A , B} …… {D}

16

The number of possible subsets of a set of size n is ?2n

A BThis is the union symbol. It means the set that consists of all elements of set A and all elements of set B.

= {1, 2, 3, 4, 5, 7, 9}Remember we do not list elements more than once.

A B

This is the intersect symbol. It means the set containing all elements that are in both A and B.

= {1, 3, 5}

A = {1, 2, 3, 4, 5} B = {1, 3, 5, 7, 9}

Union of SetsIntersection of SetsComplement of Sets

One way to represent or visualize sets is to use Venn diagrams:

Let U be the set of all students enrolled in classes this semester.

U

Let M be the set of all students enrolled in Math this semester.

Let E be the set of all students enrolled in English this semester.

U

M E

Let C be the set of all students enrolled in classes this semester, but who are not enrolled in Math or English

U

M E

C

E M = the set of students in Math AND English

U

E M

The intersection of two sets A and B, written , is the set of all members that are common to both sets.

A B

is read “A intersection B”

A B

A B

A B

Let U = and V = . 1,3,5,7 7,6,5,4

Find . U V

U V 5,7

E M = the set of students in Math OR English

U

E M

These sets can be visualized with circles in what is called a Venn Diagram.

A B

A B

Everything that is in A or B.

A B

A B

Everything that is in A AND B.

A B

The union of two sets A and B, written , is the set of all members that are common to both sets.

A B

is read “A union B”A B

A B

A B

A B

A B

Often will have a set that contains all elements that we wish to consider. This is called the universal set. All other sets are subsets of this set.

Universal Set

A B

A B = There are no elements in both A and B.When this is the case they are called disjoint sets.

AThis means the complement of A, and means the set of all elements in the universal set that are not in A.

A A

100 people were surveyed. 52 people in a survey owned a cat. 36 people owned a dog. 24 did not own a dog or cat. Draw a Venn diagram.

universal set is 100 people surveyed

C D

Set C is the cat owners and Set D is the dog owners. The sets are NOT disjoint. Some people could own both a dog and a cat.

24

Since 24 did not own a dog or cat, there must be 76 that do.

n(C D) = 76

This n means the number of elements in the set

52 + 36 = 88 so there must be 88 - 76 = 12 people that own both a dog and a cat.

1240 24

Counting Formula:

n(A B) = n(A) + n(B) - n(A B)

I would like to express my special thanks of gratitude to my teacher manish jain sir.as well as our principal sir. who gave me the golden opportunity to do this wonderful project on the topic sets, which also helped me in doing a lot of Research and i came to know about so many new things.I am really thankful to them.Secondly i would also like to thank my parents and friends who helped me a lot in finishing this project within the limited time.I am making this project not only for marks but to also increase my knowledge .THANKS AGAIN TO ALL WHO HELPED ME.

THANK YOU