Cardinality of sets

INTRODUCTION

 

What is a set?

u A set is a well defined collection of objects.

         Example:  Set with elements 1,3, 5, 7,……

                 = {1,3,5,7,..}

                 x = {x/x is an odd natural number}

u Set of real number 2x=3                 x=3/2

            A= {3/2}

             = {x/2x=3}, x is a real number

EQUALITY OF SETS

u Two sets A and B are equal if and only if every element in A belongs to B and every element in B belongs to A.

u Є              belongs to

u Example:

u A = { e,o,i,u,a }

u B = {Vowels in English alphabet}

 = { a,e,i,o,u}

       A=B

u Є            does not belong to

u Example:

            X={1,2,3}

            Y= {Natural numbers}

            x≠ Y   

            Y Є X

u CONSIDER, In an University, 64 taken Mathematics course

v 95 computer science course,

v 58 had taken physics course,

v 28 had taken Mathematics and physics course,

v 26 had taken Mathematics and computer science course,

v 22 had taken Mathematics and physics course

v 14 had taken all the three courses. Find the number of students who were surveyed. Find how many had taken one course only.

How do we find the answer for this?

Cardinal number of sets

u The number of elements in a set is called as cardinal number of sets.

            The cardinal number of a set is also called the potency of the set or power            of the set.

            Example

            A =  {1,2,3}

            B = {a,b,c}

            n(A) = 3     n(B) = 3

 

                                                                                        

 

One to one correspondence

A            B

What is the formula for solving problem involving two sets?

n(A U B) =n(A) + n(B) _ n(A ∩B)

Suppose we have three sets A, B and C and we want to find the cardinality of

A U B U C, what will be the corresponding formula?

n(F U H U C)= n(A) + n(B) + n(C) _ n(A ∩ B) – n(B ∩ C) – n(A ∩C) + n(A ∩ B ∩ C)

                             nA ∩C) +n(A ∩ B ∩ C)

A nA B U C) A) + n(B) + U C) =n(A) + n(B) + n(C) _ n(A ∩ B) – n(B ∩                   C

Solve the problem:

) –n(A C)  (A ∩ B ∩ C)

u In a group of students, 65 play foot ball, 45 play hockey, 42 play cricket, 20 play foot ball and hockey, 25 play foot ball and cricket, 15 play hockey and cricket and 8 play all the three games. Find the number of students in the groups.

 (Assume that each student in the group plays at least in one game.)

Solution:

Let F, H, and C represent the set of students who play foot ball, hockey and cricket respectively. Then

n(F) = 65,

n(H) =45 and n(C)  = 42

u Also, n(F ∩ H) = 20, n(F ∩ C) = 25, n(H ∩ C) = 15, n(F ∩ H ∩ C) = 8

         We want to find the number of students in the whole group that is

n(F U H U C).

By the formula, we have

n(F U H U C)= n(A) + n(B) + n(C) _ n(A ∩ B) – n(B ∩ C) – n(A ∩C) + n(A ∩ B ∩ C)

                                        =65+45+42-20-25-15+8

                                        = 100

         Hence the number of students in the group = 100

Alternate method

u Using venn diagram

8

25-8=7

15-8=7

20-8=12

u Number of students in the group = 28+12+18+7+10+17+8 =100

Application in  daily life

1)  A radio station surveyed 190 students to determine the types of music they liked. The survey revealed that 114 liked rock music, 50 liked folk  music, and 41 liked classical musical, 14 liked rock music and folk music, 15 liked rock music and classical music, 11 liked classical music and folk music, 5 looked all the three types of music. Find

                            i.            how many did not like any of the three types?

                          ii.            How many liked any two types only?

                       iii.            How many liked folk music but not rock music?

2) It is very much useful in the survey.) It is very much useful in the survey.

is very much useful in the survey.

∩C) + n(A ∩ B ∩ C)