SETS

Chapter-1

Sets

In mathematics set is defined as the collection of well defined object which can be separated distinctly.

For instance,

S = {2, 4, 6, 8} is collection of the even integers.

 

A set can be explained in different ways:

Listing method: A = {a, b, c, .z}

Descriptive method: N = {the natural numbers from 1 to 50}

Set builder method: A – B= A -(A∩ B )

Venn – diagram

 

 Universal sets

A universal set is the collection of all objects in a particular context or theory. All other sets in that framework constitute subsets of the universal set, which is denoted as letter U. The objects themselves are known as elements or members of U.

 

 

Subsets

The set made by elements of the universal sets is called subsets of the universal sets

For example

U = {1, 2, 3, 4, …………..50}

A = {even integers from 1 to 50}

B= {odd numbers from 1 to 50}

Here, A and B are the subsets of U

 

 

Overlapping sets

Two sets are said to be overlapped if they have same element in common.

 

 

A∩ B = {6}

A and B are overlapping sets.

 

Disjoint sets

Two sets are said to be disjoint sets if there is no element in common.

 

Cardinality of the sets

The number of the elements in the given sets is known as cardinality of sets.

A = {1, 2, 5,}

B = {5, 3, 4}

AUB = {1, 2, 3, 4, 5}

n(A) = 3

n(B) = 3

n(AUB) = 5

 

 

 

Cardinality of the three sets

LetA and B and Crepresent three sets as a shown in the figure s

 

n(AUBUC) =n(A) + n(B) + n(C) - n(A∩ B) -- n(B∩ C) - n(C∩ A) +n(A∩ B∩ C)

 

 

Finite set

A set which contains finite number of elements is called a finite set.

Eg. A = {a,b,c,d}.

 

 

Infinite set

A set which contains infinite number of elements is called an infinite set.

EgB = {x:xis a point on a line}

 

 

Singleton set

A set which contains only one element is a singleton set.

A :{ x: x set of even prime numbers}

i.eA = {2}.

 

 

Null set

A set which does not contain any element is called empty set or null set.

S = {x: x ∈ Z, x = 1/n, n ∈ N}

 N is natural number and Z is integer.

 

 

Equivalent sets

Two sets which have the same number of elements, i.e. same cardinality are equivalent sets.

P = {p. q. r, s, t} and Q = {a, e, i, o, u}

Since the two sets P and Q contain the same number of elements 5, therefore they are equivalent sets.

 

Equal sets

Two sets that contain the same elements are called equal sets.

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

 

Operation of sets

 

Union of sets

The set which includes elements of A and B is called union of the sets.

 

 

A = {1, 2, 5,}

B = {5, 3, 4}

AUB = {1, 2, 3, 4, 5}

Or, AUB = {x: xϵ A or xϵ B}

n(A) = 3

n(B) = 3

n(AUB) = n(A) + n(B) - n(A∩ B)

= 3 +3 -1 = 5

 

 

Intersections of sets

If the elements of set belongs to both Sets A and B, it is called intersection of A and B.

A = {1, 2, 5}

B = {5, 3, 4}

A∩ B = {5}

n(A∩ B) = n(A) – n₀ (A) = 3 -2 = 1

Or, A∩ B = {x: xϵ A and x ϵ B}

 

Complement of sets

The set that contains all the elements of universal sets except the given set A is called complement of the set A . It is denoted byA̅

 

Difference of sets

If A and B are the two sets , the difference of the dsets is the elements of the set thst includes only in one set .

A – B= A -(A∩ B )

B- A= B-(A∩ B )

 

Example

 

Examples1

In a group of 200 students who like game, 120 like cricket game an 105 like football game. By drawing Venn diagram find

i. how many students like both the games ?

ii. How many students like only cricket?

 

Soln

n(U) = 200

 C and F denote the students who study Cricket and football respectively.

n(C) =120

n(F) = 105

n( C ∩ F)=?

We have

n (CUF)=n(C) + n(F) -n(C∩ F)

200 = 120 + 105 -n(C∩ F)

n(C∩ F)= 25

n₀(C) = n(C) -n( C ∩ F)= 120 – 25 = 95

 

 

Examples 2

In the certain examination, 50% students passed in account, 30% passed in English, 30% failed in both and 25 student passes in both subjects. By drawing Venn – diagram, find the number of the students who passes in account only.

Let total number be x A and E denotes the students who study account and English respectively

n(A) =50%

n(E) = 30%

 = 30 %

n( A U E) = 25

We have

 n(U)= n(A)+ n(E)+${\rm{\: n\: }}\overline {\left( {{\rm{AUE}}} \right)} $ - n( A ∩E)

100%= 50 % + 30% + 30 % - n( A ∩E)

n( A ∩E) = 10%

 According to the question,

10% of x = 25

x= 250

The number of the students who passed in accounts only = 40% of the 250 = 100

 

 

Examples 3

 In a survey it was found that 8.%people like oranges ,85% like mangoes and 75% like both But 45 people like none of themDrawing Venn –diagram , find the number of the people whowere in the survey.

Soln:

Let O and M be the number of people who like oranges and mangoes respectively.

n (U) =n(OUM) + $\overline {{\rm{n}}\left( {{\rm{OUM}}} \right)} $

100%=n (O) + n (M) - n(O∩M) + $\overline {{\rm{n}}\left( {{\rm{OUM}}} \right)} $

100% =80% + 85% - 75% + $\overline {{\rm{n}}\left( {{\rm{OUM}}} \right)} $

$\overline {{\rm{n}}\left( {{\rm{OUM}}} \right)} {\rm{\: }}$= 10%

According to the question,

10%of total number(say x)= 45

Or,x=$\frac{{4500}}{{10}}$ = 450