Compiled by

Prof . Sandeep Gupta

SETS

Meaning Set is a WELL DEFINED collection of objects

Meaning

MeaningBag is a set which consists of well

definedObjects i.e. pencils, erasers, scale,books etc. These objects are the

elements ofthis set

Examples

ExamplesCan a beautiful girl or group of beautiful girls

bea set?

ExamplesThe answer is ‘NO’ because beauty is not well defined

ExamplesCan a group of boys be a collection of Set?

‘YES’ , group of boys is a set because numberof boys in the group are well defined

ExamplesA smart man …. Is it a Set?

Ans. ………

ExamplesGroup of good teachers in school…….is it acollection of set?

Ans………

ExamplesCollection of prime numbers?

Ans…….

SETSIndiviual object in the set is called ELEMENT

or a MEMBER of the set.Sets are denoted by capital alphabets , e.g. A,

B, C etc.Elements of set are denoted by small

alphabets , e.g. a, b, c etc.If a is an element of set X then we write it as

a Є XIf a is not an element of set X, then it is

written as a Є X

Types of Sets

Types of

sets

empty set

Singeltonset

Finiteset

Infiniteset

Types of SetsEmpty set : it’s a set having no element. Also

known as Null set, it is denoted by ФSingleton set : it’s a set having only one elementFinite set : it’s a set wherein counting of

elements ends at a certain stageInfinite set : it’s a set in which counting of

elements do not end at any stageAn empty set is a finite setNatural and whole numbers, integers, rational

and real numbers are infinite sets

Few concepts worth rememberingNatural and Whole numbers these includes numbers like 1,2,3,4 etconly difference is we include ‘0’ (zero) in

whole numbers but not in natural numbersare not fractions, decimals or negative

numbersIntegers are positive or negative numbers and

includes zeroare whole numbers but can be negative too

Few concepts worth rememberingRational numbers Includes integers, fractions and repeating

decimals• Irrational numbers Includes only decimals that have no pattern and

continue forever• Real numbers It includes every thing discussed above

SymbolsSymbols commonly used:Whole numbers WIntegers IRational numbersQNatural numbers NReal numbers R

Methods of writing setsListing method (Roster form)Rule method (Set builder form)Examples

1. Set of first 20 even natural numbersRoster formA = {2, 4, 6, 8, ………40}Set builder formA = {x|x is even natural number, 2 ≤ x ≤

40}

ExamplesExample 2Set of first 10 multiples of 5Roster formA = {5, 10, 15, 20, …..50}Set builder form

A = {x|x=5n, n Є N, 1 ≤ n ≤ 10}

Concept of SubsetsIf every element of set B is an element of set

A, then set B is subset of set ASubset can be proper subset or improper

subsetIf set B is subset of set A and set A contains

at least one element which is not in set B, then set B is proper subset of set A

If set A is subset of set B and if set B is subset of set A, then they are improper subsets of each other

B A is to be read as set B is proper subset of set A

Concepts of subsetsA B is to be read as set A is super set

of set BA B is to be read as set A is improper

subset of set BEvery set is subset of itselfEmpty set is subset of every set

Universal setA non-empty set of which all the sets under

consideration are the subsets of that set is called universal set

It is denoted by ‘U’

The End