2.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:

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

ii. Descriptive method: N = {the natural numbersfrom 1 to 50}

iii. Set builder method: A – B= A -(A∩ B )

iv. 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 threesets asa 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)

 

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 orxϵ 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 xϵ 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 by A̅

 

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

 

 

Exercise 2.1

Question 1:
Which of the following are sets? Justify your answer.

1.The collection of all the months of a year begining with the letter “J”.
Solution : It is well defined so it is a set.

2.The collection of ten most talented writers of India.

Solution : The collection is not well defined so it is not a set.

3.A team of eleven best cricket batsmen of the world.

Solution :We get same collection, so the collection forms a well defined set.

4.The collection of all boys in your class.

Solution : We get same collection, so the collection forms a well defined set.

5.The collection of all even integers.

Solution :The collection of all even integers gets same collection, so the collection forms a well defined set.

Question 2:
If A={0,2,4,6}, B = {3,5,7} and C = {p, q, r}then fill the appropriate symbol, ∈ or ∉ in the blanks.

1. 0…… A
2. 3…… C
3. 4…… B
4. 8…… A
5. p…… C
6. 7…… B

Solution :

1. 0 ∈ A (Since 0 is an element of A)
2. 3 ∉ C (Since 3 is not an element of C)
3. 4 ∉ B (Since 4 is not an element of B)
4. 8 ∉ A (Since 8 is not an element of A)
5. p ∈ C (Since p is an element of C)
6. 7 ∈ B (Since 7 is an element of B)

Question 3:
Express the following statements using symbols.

1. The elements V does not belong to ‘A’.
2. ‘d’ is an element of the set ‘B’.
3. ‘l’ belongs to the set of Natural numbers N.
4. ‘8’ does not belong to the set of prime numbers P

Solution :

1. X ∉ A
2. d ∈ B
3. 1 ∈ N
4. 8 ∉ P

Question 4:
State whether the following statements are hue or false. Justify your answer

1. 5 ∉ set of prime numbers
2. S = {5, 6, 7} implies 8 ∈ S.
3. -5 ∉ W where 4 ‘W’ is the set of whole numbers
4. 8/11 ∈ Z where ‘Z’ is the set of integers.

Solution :

1. False-5 can only be divided evenly by 1 or 5, so it is a prime number.
2. False-8 does not belong to the set of prime numbers S.
3. True
4. False- An integer is number with no fractional part.

Question 5:
Write the following sets in roster form.

1. B = {x: x is a natural number smaller than 6}
2. C = {x: x is a two-digit natural number such that the sum of its digits is 8}.
3. D = {x : x is a prime number which is a divisor of 60}.
4. E = {x : x is an alphabet in BETTER}.

Solution :

1. B={1,2,3,4,5}
2. C={17,26,35,44,53,62,71}
3. D={5,3}
4. E={B,E,T,R}

Question 6:
Write tine following sets in the set-builder form.

1. {3, 6, 9, 12}
2. {2,4, 8, 16, 32}
3. {5, 25, 125, 625}
4. {1 ,4, 9, 16, 25, ……..100}

Solution :

1. A={x : x is multiple of 3 and less than 13}
2. B={x : x is in power of 2xand x is less than 6}
3. C={x : x is in power of 5 and x is less than 5}
4. D={x : x is square of natural number and not greater than 10}

Question 7:
Write tine following sets in roster form.

1. A= {x : x is a natural number greater than 50 but smaller than 100}
2. B = {x : x is an integer, x2 = 4}
3. D = {x : x is a letter in the word “LOYAL”}

Solution :

1. A ={51,52,53,54,55,56,………..,98,99}
2. B ={-2,2}
3. D={L, A, O, Y}

Question 8:

Solution :

1. {1, 2, 3, 6}= {x : x is a natural number and divisor of 6}
2. {2, 3}= x : x is prime number and a divisor of 6}
3. {M, A, T, H, E, I, C, S} = {x : x is a letter of the word MATHEMATICS}
4. {1,3,5,7,9}= {x : x is an odd natural number less than10}

Exercise 2.2

Exercise 2.3

Exercise 2.4

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%

n( A U E)c  = 30%

n( A U E) = 25

We have

 n(U)= n(A)+ n(E)+n(AUE)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯$\mathrm{n}\overline{\left(\mathrm{A}\mathrm{U}\mathrm{E}\right)}$ - 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