set theory slide

8/8/2019 set theory slide

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SET: For example ,suppose one defines the term set as a welldefined collection of objects.

Collection is a aggregate of objects or things.

Aggregate is class of things.

Class is collection.

Their must be some undefined or primitive terms.

In this Chapter we start with two undefined terms.

Element and Set

Assume that the word Set is synonyms with the wordsCollection , aggregate , Class and is comprised of elements.

The words Element , Object and Member are synonyms

If a is an element of set A. Then write a A or a is in A or a ismember of A


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If a does not belongs to A then a A

It is assumed here that if A is any set and a is any

element then either a A or a A and possibilities aremutual exclusive.

Thus , one cannot say Consider the set A of somepositive integers . Because it is not sure whether 3 A or 3A.

The following are some illustration of sets :

1) The collection of Vowels in English alphabets ,This setcontains five elements , Namely A , E , I ,O ,U.

2) The collection of first five prime natural numbers is a setcontaining 2,3,5,7,113) The collection of all status in the Indian union is a set.

4) The collection of past president of the Indian union is aset


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5) The collection of cricketers in the world who were out for 99runs in a test match is a set.6) The collection of good cricket player of India is not a set.Since the term Good player is vague and it is not well defined. Similarly , Collection of good teacher in a school is a set.However , the collection of all teachers in a school is a set. In this chapter we will have frequent interaction with somesets.1)N : Set of Natural Numbers.2)Z : Set of Integers.3)Z+: Set of all Positive Integers.4)Q : Set of all Rational Numbers.

5)Q+

: Set of all Relational Positive Numbers.6) R : Set of all Real Numbers.7)R+ : Set of all Real Positive Numbers8)C : Set of all complex Numbers.


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Description of a Set :A set is often described in the following two forms one can

make use of any one of these two ways. According toconvenience.1)Roster Form or Tabular Form .In this form a set is described by listing elements , separatedby commas ,within braces {} .

Example : The set of Vowels of English alphabets may bedescribed as {a,e,i,o,u}.The set of even natural numbers can be described as{2,4,6,}.Here the dots stand for and so on.

If A is the set of all prime numbers less than A = {2,3,5,7}Note : The order in which the elements are written in a setmakes no difference. Thus , {a,e,i,o,u} and {e,a,I,o,u} denote the same set. Also therepeataion of an element has no effect.

For example {1,2,3,2} is the same as {1,2,3}


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2) Set Builder Form : In this form a set is described by a characterizing property p(x) of itselement x. In such a case the set is described by {x : p(x) holds} or ,{x | p(x) holds} ,which is read as the set of all x such that p(x) holds ,thesymbol | or : is read as such that. In other words , in order to describe a set , a variable x (say) is writteninside the brace and then after putting a colon the common property p(x)

possessed by each element of the set is written within the braces. Example :1)The set E of all even natural numbers can be written as

E = {x : x is a natural number and x = 2n for n N}Or

E = {x : x N , x = 2n , n N}OrE = {x N : x = 2n , n N}.

2)The set of all real numbers greater then 1 and less than 1 can bedescribed as {x R : -1 < x > 1 }3) The set A = {1,2,3,4,5,6,7,8} can be written as A = {x N : x 8}


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Types of Set :1 ) Empty Set :A set is said to be empty or null pr void set if it

has no element and it is denoted by . the roster method , is denoted by {} .

Example : {x R : x2 = -2 } =

{x N : 5 < x > 6 } =

The set a given by A = {x : x is an even number grater than2 } is an empty set because 2 is the only even prime number.

A set consisting of at least one element is called a nonempty or non void set.

Note : If A and B are two empty set , then x A iff , x B issatisfied because there is no element x in either A or B towhich the condition may be applied. Thus A = B. Hencethere is only one empty set and we denote it by . Thereforearticle the is used before empty set.


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2) Singleton Set :A set consisting of a single element iscalled a singleton set.

Example : The Set {5} is a singleton set . The set {x: x N and x2 = 9} is a singleton set equal to {3}

3) Finite Set :A set is called a finite set if it is either void setor its element can be listed (Counted , Labeled ) by naturalnumbers 1,2,3 and the process of listing terminates at acertain natural number n(say).

4) Cardinal Number of Finite set : The number n is the abovedefinition is called the cardinal number or order of a finiteset A and is denoted by n(A).

5) Infinite Set :A set whose elements cannot be listed by thenatural numbers 1,2,3.n for any natural number n iscalled and infinite set.


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Example : Each one of the following sets is a finite set:

1)Set of even natural numbers less than 100.

2) Set of soldiers in India army.3) Set of even prime natural numbers

4) Set of all persons on the earth.

Example : Each one of the following sets is an infinite set.

1)Set of all points in a plane.2) Set of all lines in a plane.

3){x R : 0 < x < 1 }

6) Equivalent Sets : Two finite sets A and B are equivalent if theircardinal numbers are same i.e. n(A) = n(B).

7) Equal Set : Two set A and B are said to be equal if every elementof A is a member of B and every element of B is a member of A.

If sets A and B are equal , we write A = B , we write A B when Aand B are not equal.


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8) Subsets : Let A and B be two Sets. If every element of A is anelement of B , then A is called a subset of B.

if A is a subset of B , We write A B ,which is read is as A issubset of B or A is contained in B.

Thus , A B if a A a B.

The symbol stand for Implies

If A is a subset of b , we say that B contains A or B is a super set

of A and we write B A.If A is not a subset of B, We write A B.

Obviously , every set is a subset of itself and the empty set issubset of every set.

This two subsets are called Improper subsets.A subset A of a set B is called Propersubset of B.

If A B and we write A B.

In such a case , we also say that B is a super set of A.


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If A is proper subset of B , then there exists an element x B such that xA.

It follows immediately from this definition and thedefinition of equal set that two sets A and B are equal iff A B and B A.

Thus whenever we want to prove that two sets and B are

equal , we must prove that A B and B A.Example : {1} {1,2,3} but

{1 , 4} {1,2,3}

N Z Q R C

If A is the set of all divisors of 68 and B is the set of allprime divisor of 68 ,then B is the subset of A and we write

B A


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9) Universal Set : Set that contains all sets underconsideration i.e. it is a super set of each of the given sets

such a set is called the universal set and is denoted by U.Or

A set that contains all sets in a given context iscalled the universal set.

Example :

1 )when we study two dimensional coordinate geometry thenthe set of all point in XY plane is the universal set.

2) When we are using sets containing natural numbers thenN is the universal set.

3)If A = {1,2,3} , B = {2,4,5,6} , C = {1,3,5,7} then U ={1,2,3,4,5,6,7} can be taken as universal set.


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10) Power Set : Let A be a set then the collection or family ofall subsets of A is called the power set of A and is denoted by

P(A) .P(A) = {S : S A}

The empty set and set A itself are subsets of A and aretherefore elements of P(A).

Power set of a given set is always non empty.

Example :

1)Let A = {1,2,3} then the subset of A are

P(A) = { ,{1} , {2} , {3} , {1,2} , {1,3} , {2,3} , {1,2,3}}If A is void set , then P(A) has just one element i.e.P() = {}.


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Operation on sets :

1)Union of Set : Let A and B be two sets. The union of set A

and B is the set of all those elements which belongs either toA or to B or to both A and B.

@A B = {x : x A or x B}

@x A B x A or x B

and

x A B x A and x B

From the Venn diagram , A A B , B A B

If A and B are two sets such that A B , then A B = B ifB B , then A B = A.


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Example :

1) If A = {1,2,3} B = {1,3,5,7} then A B = {1,2,3,5,7,}

2) If A = {x : X =2n +1 , n Z } B = {x : x = 2n , n Z }

A B = {x : x is an odd integer } {x : x is an eveninteger }

A

B = {x : x is an integer} = Z3) If A1 , A2,..An is a finite family of sets , then theirunion is denoted by

Ai or A1 A2 A3 .An

4) Let A ={1,2,3} , B = {3,5} , C= {4,7,8}A B C = {1,2,3,4,5,7,8,}

i= 1

n


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2) Intersection of sets : Let A and B be two sets. Theintersection of A and B is the set of all those elements that

belong to both A and B. It is denoted by A B.

@A B = {x : x A and x B}

@x A B x A and x B

and

x A B x A or x B

A B A , A B B

If A1 ,A2 ,A3An is a finite family of sets then ,their intersection is denoted by

or A1 A2A3 .. Ann

i=1


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Example :

1) If A ={1,2,3,4,5} B = {1,3,9,12} A B = {1,3}

2) A ={1,2,3,4,5,6,7} B = {2,4,6,8,10} C = {4,6,7,8,9,10,11}A B = {2,4,6}

A B C = {4,6}

3) If A = {x : x = 2n , n Z } B = {x : x = 3n , n Z}

A

B = {.-4,-2,0,2,4,6..}

{-9,-6,0,3,6}A B = {-6 , 0 , 6 , 12 ..}

= {x : x = 6n , n Z }

4) A = {x : x = 3n , n Z} , B = {x : x =4n , n Z}

x A B x =3n and x = 4n

x is multiple of 3 and 4 both

X is multiple of 12

X =12n , n Z

A B = { x : x =12n , n Z}


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If A and B are two sets ,then A B = A , If A B and A B =B , If B A.


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3)Disjoint Sets : Two sets a and B are said to be disjoint , ifA B = .If A B { ,then A and B are said to be

intersecting or overlapping sets.Example :A = {1,2,3,4,5,6} B= {7,8,9,10,11} , C = {6,8,10,12,14}

Then A and B are disjoint Sets ,While A and C areintersecting Sets .


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4) Different of Sets : Let A and B be two sets. The differenceof A and B , written A B ,is the set of all those elements of

A which do not belong to B.A B = {x : x A and x

B }

Or

A B = {XA : x B}

@x A B x A and x B.The difference B A is the set of allthose elements of B that do not

belong to A .B A = {x B : x A }


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Example :

A = {2,3,4,5,6,7} , B = {3,5,7,9,11,13}

A B = {2,4,6}

B - A = {9,11,13}


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5) Symmetric difference of two sets : Let A and B be two Sets. Thesymmetric difference of two sets A and B is set (A-B) (B A) and isdenoted by A( B.

A( B = (A-B) (B A)= {x : x A B}Example :(1)A = {1,2,3,4,5,6,7}

B = {1,3,5,6,7,8,9}

A B = {2,4} , B A = {9}@A( B = {2,4,9}

(2) A = {x R : 0 < x < 3} , B = {x R : 1 e x u 5}A B = {x R : 0 < x < 1}

B A = {x R : 3 e x u 5}A( B = {x R : 0 < x < 1 or 3 e x u 5 }


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6) Complement of set : Let U be the universal set and let Abe a set such that A U. Then complement of A with respect

to u is denoted by A or Ac or U A and defined the set of allthose elements of U. which are not in A.

A = {x U : x A}

x A x A.

N = {1,2,3,4.} , A = {2,4,6,8,} , A = {1,3,5.}

U = {1,2,3,4,5,6,7,8,9} A = {1,3,5,7,9} , A = {2,4,6,8}

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