Set concepts

transcript

Set Set ConceptsConcepts

Introduction

Shaan Education society’s Guardian college of

education

Technology based lesson

NAME OF THE STUDENT: MALTI RAI

NAME OF GUIDE: MRS NILOFER MOMIN NAME OF THE INCHARGE:MRS LEENA

CHOUDHRY

SUBJECT: Mathematics UNIT : Set concepts STANDARD: IX

1. Objectives2. Definition of set3. Properties of sets4. Set theory5. Venn Diagram6. Set Representation7. Types of Sets8. Operation on Sets

Understanding set theory helps people to …

• see things in terms of systems

• organize things into groups

• begin to understand logic

Objectives

Definition of set

• A set is a well defined collection of objects.

• Individual objects in set are called as elements of set.

e. g. 1. Collection of even numbers between 10 and 20.

2. Collection of flower or bouquet.

Properties of Sets

1 Sets are denoted by capital letters.

Set notation : A ,B, C ,D

2. Elements of set are denoted by small letters.

Element notation : a,d,f,g, For example SetA= {x,y,v,b,n,h,}

3 If x is element of A we can write as

xA i.e x belongs to set A.

4. If x is not an element of A we can write as

xA i.e x does not belong to A

e.g If Y is a set of days in a week then

Monday A and January A

5 Each element is written once.6 Set of Natural no. represented by-N,

Whole no by- W ,Integers by – I, Rational no by-Q, Real no by- R

7 Order of element is not important. i.e set A can be written as { 1,2,3,4,5,} or as {5,2,3,4,1} There is no difference between two.

Set Theory Georg cantor a German

Mathematician born in Russia is creator of set theory

The concept of infinity was developed by cantor.

Proved real no. are more numerous than natural numbers.

Defined cardinal and ordinal no.

Georg cantor

Venn Diagrams Born in 1834 in England. Devised a simple

diagramatic way to represent sets.

Here set are represented by closed figures such as :

John VennJohn Venn

.a .i.g .y

.2 .2 .6 .8.6 .8

Set Representation

• There are two main ways of representing sets.

1. Roaster method or Tabular method.

2. Set builder method or Rule method

Roster or Listing method

• All elements of the sets are listed,each element separated by comma(,) and enclosed within brackets

Roster or Listing method• All elements of the sets are

listed,each element separated by comma(,) and enclosed within brackets { }

• e.g Set C= {1,6,8,4}• Set T

={Monday,Tuesdy,Wednesday,Thursday,Friday,Saturday}

• Set k={a,e,i,o,u}

Rule method or set builder method

• All elements of set posses a common property

• e.g. set of natural numbers is represented by

• K= {x|x is a natural no}

Here | stands for ‘such that’ ‘:’ can be used in place of ‘|’

e.g. Set T={y|y is a season of the year} Set H={x|x is blood type}

Cardianility of set• Number of element in a set is called

as cardianility of set. No of elements in set n (A) e.g Set A= {he,she, it,the, you} Here no. of elements are n |A|=5

Singleton set containing only one elements e.g Set A={3}

Types of set

1. Empty set2. Finite set3. Infinite set4. Equal set5. Equivalent set6. Subset Universal set

Equal sets

• Two sets k and R are called equal if they have equal numbers and of similar types of elements.

• For e.g. If k={1,3,4,5,6}• R={1,3,4,5,6} then

both Set k and R are equal.• We can write as Set K=Set R

Empty sets

• A set which does not contain any elements is called as Empty set or Null or Void set. Denoted by or { }e.g. Set A= {set of months containing 32 days}Here n (A)= 0; hence A is an empty set.

e.g. set H={no of cars with three wheels}• Here n (H)= 0; hence it is an empty set.

Finite set

• Set which contains definite no of element.

• e.g. Set A= {,,,}• Counting of elements is fixed.Set B = { x|x is no of pages in a

particular book} Set T ={ y|y is no of seats in a bus}

Infinite set

• A set which contains indefinite numbers of elements.

Set A= { x|x is a of whole numbers}

Set B = {y|y is point on a line}

Subset• Sets which are the part of

another set are called subsets of the original set. For example, if A={3,5,6,8} and B ={1,4,9} then B is a subset of A it is represented as BA

• Every set is subset of itself i.e A A

• Empty set is a subset of every set. i.e A

.3 .5 .6.

Universal set• The universal set is the set of all

elements pertinent to a given discussionIt is designated by the symbol U

e.g. Set T ={The deck of ordinary playing cards}. Here each card is an element of universal set.

Set A= {All the face cards}Set B= {numbered cards}Set C= {Poker hands} each of these

sets are Subset of universal set T

Operation on Sets

• Intersection of sets

• Union of sets

• Difference of two sets

• Complement of a set

Intersection of sets• Let A and B be two sets. Then the

set of all common elements of A and B is called the Intersection of A and B and is denoted by A∩B

• Let A={1,2,3,7,11,13}} B={1,7,13,4,10,17}}

• Then a set C= {1,7,13}} contains the elements common to both A and B

• Hence A∩B is represented by shaded part in venn diagram.

• Thus A∩B={x|xA and xB}

Union of sets• Let A and B be two

given sets then the set of all elements which are in the set A or in the set B is called the union of two sets and is denoted by AUB and is read as ‘A union B’

•Union of Set A= {1, 2, 3, 4, = {0, 2, 4, 6}

•5, 6} and Set B

Difference of two sets1. The difference of set A- B is

set of all elements of “A” which does not belong to “B”.

2. In set builder form difference of set is:-

A-B= {x: xA xB} B-A={x: x B xA} e.g SetA ={ 1,4,7,8,9} Set B= {3,2,1,7,5} Then A-B = { 4,8,9}

Disjoint sets• Sets that have no

common members are called disjoint sets.

• Example: Given that• U=

{1,2,3,4,5,6,7,8,9,10}• setA={ 1,2,3,4,5}• setC={ 8,10}• No common elements

hence set A and are disjoint set.

Summarisation

1. Definition of set andProperties of sets

2. Set theory3. Venn Diagram4. Set Representation5. Types of Sets

Home work

1 Write definition of set concepts.2 What is intersection and union of

sets.3 Explain properties of sets with

examples.

Applications

1. A set having no element is empty set.( yes/no)2.A set having only one element is

singleton set. (yes/no)3.A set containing fixed no of elements.

{ finite/ infinite set) 4.Two set having no common element.

(disjoint set /complement set)

• Yes your answer is right

No is wrong your answer

• No your answer is wrong