The theory of sets was

developed by German

mathematician Georg Cantor

(1845-1918). He first

encountered sets while working

on “Problems on Trigonometric

Series” . SETS are being used in

mathematics problem since they

were discovered.

Collection of object of a particular kind,

such as, a pack of cards, a crowed of

people, a cricket team etc. In mathematics

of natural number, prime numbers etc.

Elements of a set aresynonymous terms.

Sets are usually denotedby capital letters.

Elements of a set arerepresented by smallletters.

There are two ways torepresent sets

Roster or tabularform.

Set-builder form.

In roster form, all theelements of set are listed,the elements are beingseparated by commas and areenclosed within braces { }.e.g. : set of

1,2,3,4,5,6,7,8,9,10.

SET-BUILDER FORM

In set-builder form, all the elements of a set possess a single common property which is not possessed by an element outside the set.e.g. : set of natural numbers k

k= {x : x is a natural number}

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

Empty sets.

Finite &Infinite sets.

Equal sets.

Subset.

Power set.

Universal set.

THE EMPTY SET

A set which doesn't contains any element is called theempty set or null set or void set, denoted bysymbol ϕ or { }.

e.g. : let R = {x : 1< x < 2, x is a naturalnumber}

A set which is empty or consistof a definite numbers ofelements is called finiteotherwise, the set is calledinfinite.e.g. : let k be the set of the daysof the week. Then k is finite

let R be the set of pointson a line. Then R is infinite

Given two sets K & r are said to be equal if they have exactly the same element and we write K=R. otherwise the sets are said to be unequal and we write K=R.e.g. : let K = {1,2,3,4} & R= {1,2,3,4}

then K=R

A set R is said to be subset of a set K if every element of R is also an element K.R ⊂ KThis mean all the elements of R contained in K

The set of all subset of a givenset is called power set of thatset.The collection of all subsets ofa set K is called the power setof denoted by P(K).In P(K)every element is a set.If K= [1,2}P(K) = {ϕ, {1}, {2}, {1,2}}

Universal set is set whichcontains all object, includingitself.e.g. : the set of real numberwould be the universal set of allother sets of number.NOTE : excluding negative root

The set of natural numbers N={1,2,3,4,....} The set of integers Z= {…,-2, -1,0, 1, 2,3,…..} The set of rational numbers Q= {x :x = p/q, p, q ∈ Z and q ≠ 0

NOTE : members of Q also includenegative numbers.

OPEN INTERVAL

The interval denoted as (a,b), a &b are real numbers ; isan open interval, meansincluding all the elementbetween a to b but excludinga &b.

The interval denoted as[a, b], a &b are Realnumbers ; is an openinterval, means includingall the element between ato b but including a &b.

(a, b) = {x : a < x < b} [a, b] = {x : a ≤ x ≤ b} [a, b) = {x : a ≤ x < b} (a, b) = {x : a < x ≤ b}

A Venn diagram or set diagram isa diagram that shows allpossible logical relations between afinite collection of sets. Venndiagrams were conceived around 1880by John Venn. They are used to teachelementary set theory, as well asillustrate simple set relationshipsin probability, logic,statistics linguistics and computer

Venn consist of rectangles and closed curves usually circles. The universal is represented usually by rectangles and its subsets by circle.

ILUSTRATION 1. in fig U= { 1, 2 , 3, ….., 10 } is the universal set of which A = { 2, 4, 3, ……, 10} is a subset.

. 2

. 4

. 8

.6

.10

. 3

. 7

. 1

. 5

. 9

ILLUSTRATION 2. In fig U = { 1, 2, 3, …., 10 } is the universal set of which A = { 2, 4, 6, 8, 10 } and B = { 4, 6 } are subsets, and also B ⊂ A . 2 A

B

. 8 . 4

. 6

. 10

. 3

. 5

.7

. 1

. 9

UNION OF SETS : the union of two sets A and B is the set C which consist of all those element which are either in A or B or in both. PURPLE part is

the union

A U B

(UNION)

1) A U B = B U A ( commutative law )2) ( A U B ) U C = A U ( B U C )

( associative law)3) A U ϕ = A ( law of identity element )4) A U A = A ( idempotent law )5) U U A = A ( law of U )

1) A ∩ B = B ∩ A ( commutative law)2) ( A ∩ B ) ∩ C = A ∩ ( B ∩ C )

( associative law)3) Φ ∩ A = Φ, U ∩ A = A ( law of Φand U )4) A ∩ A = A ( idempotent law)5) A ∩ ( B U C ) = ( A ∩ B ) U ( A ∩ C )

(

1) Let U = { 1, 2, 3, } now the set of all those element of U which doesn’t belongs to A will be called as A compliment.

U

A

A’

GREY part

shows A

complement

1) Complement laws : 1) A U A’ = U1) 2) A ∩ A’ = Φ2) 2) De Morgan’s law : 1) ( A U B )’ = A’ ∩ B’3) 2) ( A ∩ B )’ = A’ U B’4) 3) Laws of double complementation : ( A’ ) ‘ = A5) 4) Laws of empty set and universal set :

6) Φ ‘ = U & U’ = Φ