Sets

State which of the following collections are sets:

(i) collection of odd natural numbers less than 50

(ii) collection of four colours of a rainbow

(iii) collection of first three days of a week

(iv) collection of all tall students of your class

(v) collection of all lovely flowers

(vi) collection of all rich people of Bengaluru

(vii) collection of some multiples of 5

(viii) collection of all even integers which lie between -5 and 15

(ix) collection of three youngest students of your class

(x) collection of three months of a year.

Answer

(i) It is a set as it is well defined. The collection contains the odd natural numbers 1, 3, 5, 7, ......, 49.

(ii) It is not a set because it is not known which four colours of a rainbow are to be included in the collection.

(iii) It is a set as it is well defined. The collection contains the first three days of a week i.e. Sunday, Monday and Tuesday.

(iv) It is not a set because the given collection is not well defined—people may differ on whether a student of the class is tall or not.

(v) It is not a set because the given collection is not well defined—people may differ on whether a flower is lovely or not.

(vi) It is not a set because the given collection is not well defined—people may differ on whether a person of Bengaluru is rich or not.

(vii) It is not a set because it is not known which multiples of 5 are to be included in the collection.

(viii) It is a set as it is well defined. The collection contains the even integers -4, -2, 0, 2, 4, 6, 8, 10, 12, 14.

(ix) It is a set as it is well defined. The age of every student of the class can be determined and so the three youngest students of the class can be identified.

(x) It is not a set because it is not known which three months of a year are to be included in the collection.

If E = {even integers}, then insert the appropriate symbol ∈ or ∉ in the blanks:

(i) 10 .... E

(ii) -8 .... E

(iii) 13 .... E

(iv) {6} .... E

(v) a .... E

(vi) -4, 12 .... E

Answer

E = {even integers} = {......, -6, -4, -2, 0, 2, 4, 6, ......}

(i) 10 is an even integer, so it is a member of E.

10 ∈ E

(ii) -8 is an even integer, so it is a member of E.

-8 ∈ E

(iii) 13 is an odd integer, so it is not a member of E.

13 ∉ E

(iv) {6} is a set and not a member of E.

{6} ∉ E

(v) a is a letter, not an integer. So it is not a member of E.

a ∉ E

(vi) Both -4 and 12 are even integers, so both are members of E.

-4, 12 ∈ E

If V = {vowels in English alphabet}, write which of the following statements are true and which are false:

(i) c ∈ V

(ii) {a} ∈ V

(iii) a, e, i ∈ V

(iv) a, b ∈ V

(v) {a, u} ∉ V

(vi) {a, o, u} ∈ V

Answer

V = {vowels in English alphabet} = {a, e, i, o, u}

(i) False; as c is a consonant, so c ∉ V.

(ii) False; as {a} is a set and not a member of V.

(iii) True; as a, e and i are members of V.

(iv) False; as b is not a member of V i.e. b ∉ V. Of course, a ∈ V.

(v) True; as {a, u} is a set and therefore not a member of V.

(vi) False; as {a, o, u} is a set and not a member of V.

Write the following sets in roster form:

(i) the set of first five odd counting numbers

(ii) the set of all even natural numbers less than 101

(iii) {months of year whose names begin with a vowel}

(iv) {one digit natural numbers which are perfect squares}

(v) the set of multiples of 7 which lie between -20 and 25

(vi) {factors of 36}

(vii) {prime factors of 360}

(viii) the set of whole numbers which are multiples of 5

(ix) the set of all letters in the word 'CHENNAI'

(x) the set of all vowels in the word 'MUSSOORIE'

(xi) the set of all consonants in the word 'MATHEMATICS'

Answer

(i) The first five odd counting numbers are 1, 3, 5, 7 and 9.

The required set = {1, 3, 5, 7, 9}

(ii) The even natural numbers less than 101 are 2, 4, 6, ......, 100.

The required set = {2, 4, 6, ......, 100}

(iii) The months whose names begin with a vowel (a, e, i, o, u) are April, August and October.

The required set = {April, August, October}

(iv) The one digit natural numbers are 1, 2, 3, 4, 5, 6, 7, 8 and 9. Out of these, the perfect squares are 1, 4 and 9.

The required set = {1, 4, 9}

(v) The multiples of 7 are ......, -21, -14, -7, 0, 7, 14, 21, 28, ...... The multiples of 7 lying between -20 and 25 are -14, -7, 0, 7, 14 and 21.

The required set = {-14, -7, 0, 7, 14, 21}

(vi) The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.

The required set = {1, 2, 3, 4, 6, 9, 12, 18, 36}

(vii) 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5

So, the prime factors of 360 are 2, 3 and 5.

The required set = {2, 3, 5}

(viii) The whole numbers which are multiples of 5 are 0, 5, 10, 15, 20, ......

The required set = {0, 5, 10, 15, 20, ......}

(ix) The letters in the word 'CHENNAI' are C, H, E, N, N, A, I. Writing each element only once, we get C, H, E, N, A, I.

The required set = {C, H, E, N, A, I}

(x) The vowels in the word 'MUSSOORIE' are U, O, O, I, E. Writing each element only once, we get U, O, I, E.

The required set = {U, O, I, E}

(xi) The letters in the word 'MATHEMATICS' are M, A, T, H, E, M, A, T, I, C, S. The consonants among these are M, T, H, M, T, C, S. Writing each element only once, we get M, T, H, C, S.

The required set = {M, T, H, C, S}

Write the following sets in tabular form:

(i) {x : x is a natural number and x < 7}

(ii) {x : x ∈ W and x ≤ 5}

(iii) {x : x is a month of a year having less than 31 days}

(iv) {x | x is a letter in the word 'CIRCUMFERENCE'}

(v) {x | x is a vowel in the word 'NOTATION'}

(vi) {x : x is a digit in the numeral 110526715}

(vii) {x : x is a factor of 48}

(viii) {x : x is a multiple of 11 and 0 ≤ x < 80}

(ix) {y : y is a two digit natural number divisible by 10}

Answer

(i) The natural numbers less than 7 are 1, 2, 3, 4, 5 and 6.

{x : x is a natural number and x < 7} = {1, 2, 3, 4, 5, 6}

(ii) The whole numbers less than or equal to 5 are 0, 1, 2, 3, 4 and 5.

{x : x ∈ W and x ≤ 5} = {0, 1, 2, 3, 4, 5}

(iii) The months of a year having less than 31 days are February (28 or 29 days), April (30 days), June (30 days), September (30 days) and November (30 days).

{x : x is a month of a year having less than 31 days} = {February, April, June, September, November}

(iv) The letters in the word 'CIRCUMFERENCE' are C, I, R, C, U, M, F, E, R, E, N, C, E. Writing each letter only once, we get C, I, R, U, M, F, E, N.

{x | x is a letter in the word 'CIRCUMFERENCE'} = {C, I, R, U, M, F, E, N}

(v) The letters in the word 'NOTATION' are N, O, T, A, T, I, O, N. The vowels among these are O, A, I.

{x | x is a vowel in the word 'NOTATION'} = {O, A, I}

(vi) The digits in the numeral 110526715 are 1, 1, 0, 5, 2, 6, 7, 1, 5. Writing each digit only once, we get 1, 0, 5, 2, 6, 7.

{x : x is a digit in the numeral 110526715} = {1, 0, 5, 2, 6, 7}

(vii) The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48.

{x : x is a factor of 48} = {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}

(viii) The multiples of 11 are 0, 11, 22, 33, 44, 55, 66, 77, 88, ...... Out of these, those satisfying 0 ≤ x < 80 are 0, 11, 22, 33, 44, 55, 66 and 77.

{x : x is a multiple of 11 and 0 ≤ x < 80} = {0, 11, 22, 33, 44, 55, 66, 77}

(ix) The two digit natural numbers divisible by 10 are 10, 20, 30, 40, 50, 60, 70, 80 and 90.

{y : y is a two digit natural number divisible by 10} = {10, 20, 30, 40, 50, 60, 70, 80, 90}

Write the following sets in roster form and also in set builder form:

(i) the set of integers which lie between -2 and 3 (both inclusive)

(ii) the set of letters in the word 'ULTIMATUM'

(iii) {months of a year whose names begin with J}

(iv) the set of single digit whole numbers which are perfect squares

Answer

(i) The integers between -2 and 3 (both inclusive) are -2, -1, 0, 1, 2 and 3.

Roster form: {-2, -1, 0, 1, 2, 3}

Set builder form: {x : x ∈ I and -2 ≤ x ≤ 3}

(ii) The letters in the word 'ULTIMATUM' are U, L, T, I, M, A, T, U, M. Writing each letter only once, we get U, L, T, I, M, A.

Roster form: {U, L, T, I, M, A}

Set builder form: {x : x is a letter in the word 'ULTIMATUM'}

(iii) The months of a year whose names begin with J are January, June and July.

Roster form: {January, June, July}

Set builder form: {x : x is a month of a year whose name begins with J}

(iv) The single digit whole numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Out of these, the perfect squares are 0, 1, 4 and 9.

Roster form: {0, 1, 4, 9}

Set builder form: {x : x is a one-digit whole number and a perfect square}

State whether the following sets are empty, finite or infinite sets. In case of (non-empty) finite sets, mention the cardinal number.

(i) {all colours of a rainbow}

(ii) {x | x is a prime number between 7 and 11}

(iii) {x | x is a digit in the numeral 550131527}

(iv) {x | x is a letter in the word 'SUFFICIENT'}

(v) {x | x is a vowel in the word MATHEMATICS}

(vi) {x : x is an even whole number and x ≤ 20}

(vii) {x : x ∈ I and -2 ≤ x < 5}

(viii) {x : x is a prime number less than 25}

(ix) {x : x is a prime factor of 180}

(x) {x : x ∈ N and x is a composite number < 12}

Answer

(i) The colours of a rainbow are Violet, Indigo, Blue, Green, Yellow, Orange and Red.

{all colours of a rainbow} = {Violet, Indigo, Blue, Green, Yellow, Orange, Red}

It has 7 elements.

Hence, it is a finite set with cardinal number 7.

(ii) There is no prime number between 7 and 11 (since 8, 9 and 10 are not prime).

{x | x is a prime number between 7 and 11} = { }

Hence, it is an empty set.

(iii) The digits in the numeral 550131527 are 5, 5, 0, 1, 3, 1, 5, 2, 7. Writing each digit only once, we get 5, 0, 1, 3, 2, 7.

{x | x is a digit in the numeral 550131527} = {5, 0, 1, 3, 2, 7}

It has 6 elements.

Hence, it is a finite set with cardinal number 6.

(iv) The letters in the word 'SUFFICIENT' are S, U, F, F, I, C, I, E, N, T. Writing each letter only once, we get S, U, F, I, C, E, N, T.

{x | x is a letter in the word 'SUFFICIENT'} = {S, U, F, I, C, E, N, T}

It has 8 elements.

Hence, it is a finite set with cardinal number 8.

(v) The letters in the word MATHEMATICS are M, A, T, H, E, M, A, T, I, C, S. The vowels among these are A, E, A, I. Writing each letter only once, we get A, E, I.

{x | x is a vowel in the word MATHEMATICS} = {A, E, I}

It has 3 elements.

Hence, it is a finite set with cardinal number 3.

(vi) The even whole numbers ≤ 20 are 0, 2, 4, 6, 8, 10, 12, 14, 16, 18 and 20.

{x : x is an even whole number and x ≤ 20} = {0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20}

It has 11 elements.

Hence, it is a finite set with cardinal number 11.

(vii) The integers x such that -2 ≤ x < 5 are -2, -1, 0, 1, 2, 3 and 4.

{x : x ∈ I and -2 ≤ x < 5} = {-2, -1, 0, 1, 2, 3, 4}

It has 7 elements.

Hence, it is a finite set with cardinal number 7.

(viii) The prime numbers less than 25 are 2, 3, 5, 7, 11, 13, 17, 19 and 23.

{x : x is a prime number less than 25} = {2, 3, 5, 7, 11, 13, 17, 19, 23}

It has 9 elements.

Hence, it is a finite set with cardinal number 9.

(ix) 180 = 2 × 2 × 3 × 3 × 5 = 2² × 3² × 5

So, the prime factors of 180 are 2, 3 and 5.

{x : x is a prime factor of 180} = {2, 3, 5}

It has 3 elements.

Hence, it is a finite set with cardinal number 3.

(x) The composite natural numbers less than 12 are 4, 6, 8, 9 and 10.

{x : x ∈ N and x is a composite number < 12} = {4, 6, 8, 9, 10}

It has 5 elements.

Hence, it is a finite set with cardinal number 5.

State whether the following pairs of sets are equal or not:

(i) A = {2, 4, 6, 8, 10}, B = {even natural numbers}

(ii) A = {3, 5, 7, 9, 11, 13}, B = {odd natural numbers between 2 and 14}

(iii) A = {PUPPET}, B = {P, U, E, T}

(iv) A = {x | x is a letter in the word SOPHIA}, B = {x | x is a letter in the word MUMTAZ}

(v) A = {kids 5 metres tall}, B = {x : x ∈ N and 2x = 3}

Answer

(i) A = {2, 4, 6, 8, 10}

B = {even natural numbers} = {2, 4, 6, 8, 10, 12, 14, ......}

The elements of A and B are not the same as B contains many more elements than A.

Hence, A ≠ B.

(ii) A = {3, 5, 7, 9, 11, 13}

The odd natural numbers between 2 and 14 are 3, 5, 7, 9, 11 and 13.

B = {odd natural numbers between 2 and 14} = {3, 5, 7, 9, 11, 13}

The elements of A and B are exactly the same.

Hence, A = B.

(iii) A = {PUPPET}

This set has only one element, namely the word "PUPPET".

B = {P, U, E, T}

This set has four elements P, U, E and T.

The elements of A and B are not the same.

Hence, A ≠ B.

(iv) A = {x | x is a letter in the word SOPHIA} = {S, O, P, H, I, A}

B = {x | x is a letter in the word MUMTAZ} = {M, U, T, A, Z}

The elements of A and B are not the same.

Hence, A ≠ B.

(v) A = {kids 5 metres tall}

There is no kid who is 5 metres tall, so A = { } i.e. A is an empty set.

B = {x : x ∈ N and 2x = 3}

2x = 3 ⇒ x = $\dfrac{3}{2}$, which is not a natural number.

So, B = { } i.e. B is also an empty set.

Both A and B have no elements.

Hence, A = B.

Given that A = {2, 5, 7, 8, 10}, B = {5, 7, 2, x, 10} and A = B, write the value of x.

Answer

A = {2, 5, 7, 8, 10}

B = {5, 7, 2, x, 10}

Since A = B, both sets must have exactly the same elements.

The elements 5, 7, 2 and 10 are common to both sets.

The remaining element in A is 8, which must also be present in B.

So, x = 8.

Hence, x = 8.

Fill in the blanks:

(i) A collection of ... objects is called a set.

(ii) If x is a member of the set A, we write it as ... .

(iii) The order of listing the elements of a set can be ... .

(iv) If one or more elements are repeated, the set remains ... .

Answer

(i) A collection of well defined objects is called a set.

(ii) If x is a member of the set A, we write it as x ∈ A.

(iii) The order of listing the elements of a set can be changed.

(iv) If one or more elements are repeated, the set remains the same.

State whether the following statements are true (T) or false (F). Justify your answer.

(i) A collection of stamps is a set.

(ii) A collection of some fruits is a set.

(iii) A group of boys playing cricket is a set.

(iv) Collection of five rivers of India is a set.

Answer

(i) False; it is not a set because it is not known which stamps are included in the collection.

(ii) False; it is not a set because it is not known which fruits are included in the collection.

(iii) False; it is not a set because it is not known which students are included in the group.

(iv) False; it is not a set because it is not known which five rivers of India are included in the collection.

Which of the following collections is a set?

1. Collection of all tasty fruits

2. Collection of all good football players of your school

3. Collection of all months of a year

4. Collection of 5 most intelligent students of your class.

Answer

Checking all options :

1. Collection of all tasty fruits — Not a set, since 'tasty' is subjective.

2. Collection of all good football players of your school — Not a set, since 'good' is subjective.

3. Collection of all months of a year — This is a set, since the months of a year are well defined: January, February, ......, December.

4. Collection of 5 most intelligent students of your class — Not a set, since 'most intelligent' is subjective.

Hence, option 3 is the correct option.

The method of representation used in the set A = {x | x is an even natural number less than 15} is called

1. Description method

2. Rule method

3. Roster method

4. none of these

Answer

In set A = {x | x is an even natural number less than 15}, a variable x is written followed by a property satisfied by each member of the set, enclosed in curly brackets. This method is called the rule method (or set builder form).

Hence, option 2 is the correct option.

The cardinal number of the empty set is

1. 2

2. 1

3. 0

4. none of these

Answer

The empty set has no elements. So, its cardinal number is 0 i.e. n(φ) = 0.

Hence, option 3 is the correct option.

If S = {x | x is a letter in the word AHMEDABAD}, then the cardinal number of S is

1. 9

2. 8

3. 7

4. 6

Answer

The letters in the word AHMEDABAD are A, H, M, E, D, A, B, A, D. Writing each letter only once, we get A, H, M, E, D, B.

S = {A, H, M, E, D, B}

So, the cardinal number of S = 6 i.e. n(S) = 6.

Hence, option 4 is the correct option.

If A = {x : x ∈ N and x is an odd prime number less than 17}, then the cardinal number of A is

1. 8

2. 6

3. 5

4. none of these

Answer

The prime numbers less than 17 are 2, 3, 5, 7, 11 and 13. Out of these, the odd primes are 3, 5, 7, 11 and 13.

A = {3, 5, 7, 11, 13}

So, the cardinal number of A = 5 i.e. n(A) = 5.

Hence, option 3 is the correct option.

{months of a year whose names begin with the letter F} is

1. an infinite set

2. empty set

3. singleton set

4. none of these

Answer

The only month of a year whose name begins with the letter F is February.

{months of a year whose names begin with the letter F} = {February}

This set has only one element, so it is a singleton set.

Hence, option 3 is the correct option.

Statement I: {2, 3, 4} and {4, 3, 2} are the same sets.

Statement II: In a set, the order of writing the elements does not matter.

1. Statement I is true but statement II is false.

2. Statement I is false but statement II is true.

3. Both Statement I and statement II are true.

4. Both Statement I and statement II are false.

Answer

Statement I: {2, 3, 4} and {4, 3, 2} contain exactly the same elements 2, 3 and 4. So, they are the same sets. Hence, Statement I is true.

Statement II: The order of listing the elements of a set can be changed without affecting the set. So, Statement II is true and it correctly explains Statement I.

Hence, option 3 is the correct option.

Statement I: An empty set, a singleton set and a set of prime numbers are all finite sets.

Statement II: The cardinal number of a set is always non-negative.

1. Statement I is true but statement II is false.

2. Statement I is false but statement II is true.

3. Both Statement I and statement II are true.

4. Both Statement I and statement II are false.

Answer

Statement I: The empty set has 0 elements (finite) and a singleton set has 1 element (finite). However, the set of all prime numbers is {2, 3, 5, 7, 11, 13, ......}, which is an infinite set. So, Statement I is false.

Statement II: The cardinal number of a finite set is the number of different elements in the set, which is always 0 or a positive integer i.e. it is always non-negative. So, Statement II is true.

Hence, option 2 is the correct option.

Statement I: An empty set is always a finite set but a finite set may or may not be empty.

Statement II: The cardinal number of a set of the English alphabet is less than 30.

1. Statement I is true but statement II is false.

2. Statement I is false but statement II is true.

3. Both Statement I and statement II are true.

4. Both Statement I and statement II are false.

Answer

Statement I: An empty set has 0 elements (a definite number), so it is a finite set. A finite set has a definite number of elements, which may be 0 (empty) or a positive integer (non-empty). So, Statement I is true.

Statement II: The English alphabet has 26 letters, so the cardinal number of a set of the English alphabet is 26, which is less than 30. So, Statement II is true.

Hence, option 3 is the correct option.

Statement I: The cardinal number of the set A = {x | x ∈ N, x² < 26} is 6.

Statement II: The number of different elements in a finite set is called its cardinal number.

1. Statement I is true but statement II is false.

2. Statement I is false but statement II is true.

3. Both Statement I and statement II are true.

4. Both Statement I and statement II are false.

Answer

Statement I: A = {x | x ∈ N, x² < 26}

For x = 1, x² = 1 < 26 ✓

For x = 2, x² = 4 < 26 ✓

For x = 3, x² = 9 < 26 ✓

For x = 4, x² = 16 < 26 ✓

For x = 5, x² = 25 < 26 ✓

For x = 6, x² = 36 > 26 ✗

So, A = {1, 2, 3, 4, 5} which has 5 elements i.e. n(A) = 5, not 6. So, Statement I is false.

Statement II: By definition, the number of different elements in a finite set is called its cardinal number. So, Statement II is true.

Hence, option 2 is the correct option.

State which of the given collections are sets:

(i) collection of all poor people of Dhanbad

(ii) collection of all difficult problems in your maths book

(iii) collection of all popular cinema actors of India

(iv) collection of all countries of Asia

(v) collection of four countries of Asia

(vi) collection of three cities of India whose name start with the letter 'J'

(vii) collection of all people in this world over 50 years of age.

Answer

(i) It is not a set because the given collection is not well defined—people may differ on whether a person of Dhanbad is poor or not.

(ii) It is not a set because the given collection is not well defined—people may differ on whether a problem is difficult or not.

(iii) It is not a set because the given collection is not well defined—people may differ on whether a cinema actor is popular or not.

(iv) It is a set as it is well defined. The countries of Asia can be clearly listed.

(v) It is not a set because it is not known which four countries of Asia are to be included in the collection.

(vi) It is not a set because it is not known which three cities of India whose name start with 'J' are to be included in the collection (there are many such cities like Jaipur, Jodhpur, Jabalpur, Jaisalmer, etc.).

(vii) It is a set as it is well defined. The age of every person can be determined and thus people over 50 years can be clearly identified.

If A = {3, 5, 7, 9, 11}, then write which of the following statements are true. If a statement is not true, mention why.

(i) 3 ∈ A

(ii) 5, 9 ∈ A

(iii) 8 ∉ A

(iv) 7 ∉ A

(v) {3} ∈ A

(vi) {5, 9} ∈ A

Answer

(i) True; as 3 is a member of A.

(ii) True; as 5 and 9 are both members of A.

(iii) True; as 8 is not a member of A.

(iv) Not true; because 7 ∈ A.

(v) Not true; because {3} is a set and not an element.

(vi) Not true; as {5, 9} is a set and not an element of A.

Write the following sets in the roster form:

(i) A = {x | x is a month of a year having 30 days}

(ii) B = {x | x = 2n, n ∈ W and n < 5}

(iii) C = {x | x ∈ N and x² < 40}

(iv) D = {all letters in the word PERMISSION}

(v) E = {x : x ∈ I and x² < 10}

(vi) F = {x : x ∈ N, 15 < x < 50 and x is divisible by 6}

(vii) the set of whole numbers which are greater than 14 and divisible by 7

(viii) the set of signs of four fundamental operations of arithmetic.

Answer

(i) The months of a year having 30 days are April, June, September and November.

A = {April, June, September, November}

(ii) B = {x | x = 2n, n ∈ W and n < 5}

The whole numbers less than 5 are 0, 1, 2, 3 and 4.

When n = 0, x = 2 × 0 = 0

When n = 1, x = 2 × 1 = 2

When n = 2, x = 2 × 2 = 4

When n = 3, x = 2 × 3 = 6

When n = 4, x = 2 × 4 = 8

B = {0, 2, 4, 6, 8}

(iii) C = {x | x ∈ N and x² < 40}

For x = 1, x² = 1 < 40 ✓

For x = 2, x² = 4 < 40 ✓

For x = 3, x² = 9 < 40 ✓

For x = 4, x² = 16 < 40 ✓

For x = 5, x² = 25 < 40 ✓

For x = 6, x² = 36 < 40 ✓

For x = 7, x² = 49 > 40 ✗

C = {1, 2, 3, 4, 5, 6}

(iv) The letters in the word PERMISSION are P, E, R, M, I, S, S, I, O, N. Writing each letter only once, we get P, E, R, M, I, S, O, N.

D = {P, E, R, M, I, S, O, N}

(v) E = {x : x ∈ I and x² < 10}

For x = 0, x² = 0 < 10 ✓

For x = ±1, x² = 1 < 10 ✓

For x = ±2, x² = 4 < 10 ✓

For x = ±3, x² = 9 < 10 ✓

For x = ±4, x² = 16 > 10 ✗

E = {-3, -2, -1, 0, 1, 2, 3}

(vi) F = {x : x ∈ N, 15 < x < 50 and x is divisible by 6}

The natural numbers between 15 and 50 that are divisible by 6 are 18, 24, 30, 36, 42 and 48.

F = {18, 24, 30, 36, 42, 48}

(vii) The whole numbers greater than 14 and divisible by 7 are 21, 28, 35, 42, 49, ......

The required set = {21, 28, 35, 42, 49, ......}

(viii) The four fundamental operations of arithmetic are addition, subtraction, multiplication and division. Their signs are +, -, × and ÷ respectively.

The required set = {+, -, ×, ÷}

Write the following sets in set builder form:

(i) A = {2, 3, 5, 7, 11, 13, 17, 19}

(ii) B = {all months of a year}

(iii) C = {Monday, Tuesday, Wednesday}

Answer

(i) A = {2, 3, 5, 7, 11, 13, 17, 19}

The elements of A are prime numbers less than 20.

A = {x : x is a prime number and x < 20}

(ii) B = {all months of a year}

B = {x : x is a month of a year}

(iii) C = {Monday, Tuesday, Wednesday}

The elements of C are the first three days of a week.

C = {x : x is one of the first three days of a week}

Write the following sets in roster form and also in set builder form:

(i) A = {even whole numbers which are less than 50}

(ii) B = {two digit numbers which are perfect square}

(iii) the set of letters in the word MUSSOORIE

Answer

(i) The even whole numbers less than 50 are 0, 2, 4, 6, ......, 48.

Roster form: A = {0, 2, 4, 6, ......, 48}

Set builder form: A = {x : x ∈ W and x is an even number < 50}

(ii) The two digit numbers are 10, 11, 12, ......, 99. Out of these, the perfect squares are:

4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64 and 9² = 81

Roster form: B = {16, 25, 36, 49, 64, 81}

Set builder form: B = {x : x is a perfect square and two digit number}

(iii) The letters in the word MUSSOORIE are M, U, S, S, O, O, R, I, E. Writing each letter only once, we get M, U, S, O, R, I, E.

Roster form: {M, U, S, O, R, I, E}

Set builder form: {x : x is a letter in the word MUSSOORIE}

Classify the following sets as empty set, finite set or infinite set:

(i) the set of all even prime numbers > 2

(ii) the set of even prime numbers

(iii) the set of prime numbers less than one crore

(iv) {all points on a line segment of length 3 cm}

Answer

(i) The only even prime number is 2. So, there is no even prime number greater than 2.

The set = { }

Hence, it is an empty set.

(ii) The only even prime number is 2.

The set = {2}

It has only one element.

Hence, it is a finite set (having only one element).

(iii) The prime numbers less than one crore are 2, 3, 5, 7, 11, ......, and there is a definite (countable) number of them.

Hence, it is a finite set.

(iv) A line segment, however small, contains an unlimited (uncountable) number of points.

Hence, it is an infinite set.

Find the cardinal number of the following sets:

(i) A = {x | x is a consonant in the word HUNDRED}

(ii) B = {x | x is a vowel in the word DEHRADOON}

(iii) C = {x | x ∈ W and x² < 50}

(iv) D = {x | x ∈ N and x < 1}

(v) E = {x | x is a prime number between 8 and 30}

Answer

(i) The letters in the word HUNDRED are H, U, N, D, R, E, D. The consonants among these are H, N, D, R, D. Writing each letter only once, we get H, N, D, R.

A = {H, N, D, R}, which has 4 elements.

∴ n(A) = 4.

(ii) The letters in the word DEHRADOON are D, E, H, R, A, D, O, O, N. The vowels among these are E, A, O, O. Writing each letter only once, we get E, A, O.

B = {E, A, O}, which has 3 elements.

∴ n(B) = 3.

(iii) C = {x | x ∈ W and x² < 50}

For x = 0, x² = 0 < 50 ✓

For x = 1, x² = 1 < 50 ✓

For x = 2, x² = 4 < 50 ✓

For x = 3, x² = 9 < 50 ✓

For x = 4, x² = 16 < 50 ✓

For x = 5, x² = 25 < 50 ✓

For x = 6, x² = 36 < 50 ✓

For x = 7, x² = 49 < 50 ✓

For x = 8, x² = 64 > 50 ✗

C = {0, 1, 2, 3, 4, 5, 6, 7}, which has 8 elements.

∴ n(C) = 8.

(iv) D = {x | x ∈ N and x < 1}

The natural numbers are 1, 2, 3, ...... There is no natural number less than 1.

D = { }, which has 0 elements.

∴ n(D) = 0.

(v) The prime numbers between 8 and 30 are 11, 13, 17, 19, 23 and 29.

E = {11, 13, 17, 19, 23, 29}, which has 6 elements.

∴ n(E) = 6.