Sets — Exercise 1.1

State which of the given collection of objects is a set:

(i) A collection of popular cinema actors of India.

(ii) The collection of even natural numbers less than 51.

(iii) The collection of counting numbers less than 1.

(iv) Collection of interesting books written by Shakespeare.

(v) The collection of novels written by Munshi Prem Chand.

(vi) The collection of 10 most talented students of your school.

(vii) Collection of all rivers flowing in India.

(viii) Collection of 5 rivers flowing in India.

(ix) Collection of all rational numbers which lie between −1 and 1.

(x) A team of eleven best cricketers of the world.

(xi) A collection of most dangerous animals of the world.

Answer

Any well-defined collection of objects is called a set. By 'well-defined collection of objects' we mean that given a collection and an object, it should be possible to decide beyond doubt whether the object belongs to the given collection or not.

(i) A collection of popular cinema actors of India:

It is not a well-defined collection, since an actor considered popular by one person may not be considered popular by another.

Hence, this collection is not a set.

(ii) The collection of even natural numbers less than 51:

Given any object, we can decide beyond doubt whether it is an even natural number less than 51 or not. The collection is 2, 4, 6, ....., 50.

Hence, this collection is a set.

(iii) The collection of counting numbers less than 1:

The counting numbers are 1, 2, 3, ..... and we can decide beyond doubt that none of them is less than 1. So the collection is well-defined, though it contains no object at all.

Hence, this collection is a set (it is the empty set).

(iv) Collection of interesting books written by Shakespeare:

It is not a well-defined collection, since a book considered interesting by one person may not be considered interesting by another.

Hence, this collection is not a set.

(v) The collection of novels written by Munshi Prem Chand:

Given any novel, we can decide beyond doubt whether it was written by Munshi Prem Chand or not.

Hence, this collection is a set.

(vi) The collection of 10 most talented students of your school:

It is not a well-defined collection, since a student considered talented by one person may not be considered talented by another. So the 10 students cannot be fixed beyond doubt.

Hence, this collection is not a set.

(vii) Collection of all rivers flowing in India:

Given any river, we can decide beyond doubt whether it flows in India or not.

Hence, this collection is a set.

(viii) Collection of 5 rivers flowing in India:

Many rivers flow in India and it is not stated which 5 of them are to be taken. So, given a river, we cannot decide whether it belongs to this collection or not.

Hence, this collection is not a set.

(ix) Collection of all rational numbers which lie between −1 and 1:

Given any number, we can decide beyond doubt whether it is rational and whether it lies between −1 and 1.

Hence, this collection is a set.

(x) A team of eleven best cricketers of the world:

It is not a well-defined collection, since a cricketer considered best by one person may not be considered best by another.

Hence, this collection is not a set.

(xi) A collection of most dangerous animals of the world:

It is not a well-defined collection, since an animal considered most dangerous by one person may not be considered most dangerous by another.

Hence, this collection is not a set.

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) 9 ∉ A

(v) {3} ∈ A

(vi) {5, 7} ∈ A.

Answer

Given,

A = {3, 5, 7, 9, 11}

The elements of A are the five numbers 3, 5, 7, 9 and 11.

(i) 3 ∈ A

The element 3 belongs to the set A.

Hence, the statement 3 ∈ A is true.

(ii) 5, 9 ∈ A

Both the elements 5 and 9 belong to the set A.

Hence, the statement 5, 9 ∈ A is true.

(iii) 8 ∉ A

The number 8 is not one of the elements 3, 5, 7, 9, 11 of A, so 8 does not belong to A.

Hence, the statement 8 ∉ A is true.

(iv) 9 ∉ A

The number 9 is one of the elements of A, so 9 belongs to A i.e. 9 ∈ A.

Hence, the statement 9 ∉ A is not true, because 9 ∈ A.

(v) {3} ∈ A

Here {3} is a set having one element 3. The elements of A are the numbers 3, 5, 7, 9, 11 and not the set {3}.

Hence, the statement {3} ∈ A is not true, because {3} is a set and not an element of A.

(vi) {5, 7} ∈ A

Here {5, 7} is a set having two elements 5 and 7. The elements of A are numbers, not sets, so the set {5, 7} is not an element of A.

Hence, the statement {5, 7} ∈ A is not true, because {5, 7} is a set and not an element of A.

Use the roster method to represent the set:

The counting numbers which are multiples of 6 and less than 50.

Answer

The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, ..... Out of these, the ones less than 50 are 6, 12, 18, 24, 30, 36, 42 and 48.

Hence, the required set in roster form is {6, 12, 18, 24, 30, 36, 42, 48}.

Use the roster method to represent the set:

The fractions whose numerator is 1, and whose denominator is a counting number less than 10.

Answer

The counting numbers less than 10 are 1, 2, 3, 4, 5, 6, 7, 8, 9. Taking each of these as the denominator with numerator 1, we get the fractions $\dfrac{1}{1}, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}, \dfrac{1}{5}, \dfrac{1}{6}, \dfrac{1}{7}, \dfrac{1}{8}, \dfrac{1}{9}$.

Hence, the required set in roster form is $\bm{\bigg\lbrace1, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}, \dfrac{1}{5}, \dfrac{1}{6}, \dfrac{1}{7}, \dfrac{1}{8}, \dfrac{1}{9}\bigg\rbrace}$.

Use the roster method to represent the set:

{x : x ∈ N and x is a prime factor of 84}.

Answer

Resolving 84 into prime factors,

84 = 2 × 2 × 3 × 7 = 2² × 3 × 7

∴ The prime factors of 84 are 2, 3 and 7.

Hence, the required set in roster form is {2, 3, 7}.

Use the roster method to represent the set:

The set of odd integers lying between −4 and 8.

Answer

The integers lying between −4 and 8 are −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, 7. Out of these the odd ones are −3, −1, 1, 3, 5 and 7.

Hence, the required set in roster form is {−3, −1, 1, 3, 5, 7}.

Use the roster method to represent the set:

The set of all natural numbers x for which x + 6 is less than 10.

Answer

Here x ∈ N and x + 6 < 10

⇒ x < 10 − 6

⇒ x < 4

As x is a natural number less than 4, x = 1, 2, 3.

Hence, the required set in roster form is {1, 2, 3}.

Use the roster method to represent the set:

The set of all integers x for which x + 6 is greater than 10.

Answer

Here x is an integer and x + 6 > 10

⇒ x > 10 − 6

⇒ x > 4

As x is an integer greater than 4, x = 5, 6, 7, .....

Hence, the required set in roster form is {5, 6, 7, .....}.

Use the roster method to represent the set:

The set of all integers x for which x + 6 is less than 10.

Answer

Here x is an integer and x + 6 < 10

⇒ x < 10 − 6

⇒ x < 4

As x is an integer less than 4, x = 3, 2, 1, 0, −1, −2, .....

Hence, the required set in roster form is {....., −2, −1, 0, 1, 2, 3}.

Use the roster method to represent the set:

The set of all integers x for which $\dfrac{60}{x}$ is a natural number.

Answer

$\dfrac{60}{x}$ is a natural number only when x is a positive divisor of 60.

Resolving 60 into prime factors, 60 = 2² × 3 × 5, so the positive divisors of 60 are

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

Hence, the required set in roster form is {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}.

Use the roster method to represent the set:

{3n + 5; n ∈ N and n ≤ 6}.

Answer

As n ∈ N and n ≤ 6, n = 1, 2, 3, 4, 5, 6.

Putting these values in 3n + 5, we get

⇒ 3(1) + 5 = 8,

⇒ 3(2) + 5 = 11,

⇒ 3(3) + 5 = 14,

⇒ 3(4) + 5 = 17,

⇒ 3(5) + 5 = 20,

⇒ 3(6) + 5 = 23.

Hence, the required set in roster form is {8, 11, 14, 17, 20, 23}.

Use the roster method to represent the set:

{x : x ∈ N and 4x − 3 ≤ 15}.

Answer

Here x ∈ N and 4x − 3 ≤ 15

⇒ 4x ≤ 15 + 3

⇒ 4x ≤ 18

⇒ x ≤ $\dfrac{18}{4}$

⇒ x ≤ 4.5

As x is a natural number not exceeding 4.5, x = 1, 2, 3, 4.

Hence, the required set in roster form is {1, 2, 3, 4}.

Use the roster method to represent the set:

{x : x ∈ N, x² < 40}

Answer

Here x ∈ N and x² < 40.

The squares of the natural numbers are 1, 4, 9, 16, 25, 36, 49, ..... Out of these, 1, 4, 9, 16, 25 and 36 are less than 40, and they are the squares of 1, 2, 3, 4, 5 and 6.

Hence, the required set in roster form is {1, 2, 3, 4, 5, 6}.

Use the roster method to represent the set:

{x : x ∈ Z and x² < 16}.

Answer

Here x ∈ Z and x² < 16.

The squares of the integers 0, ±1, ±2, ±3 are 0, 1, 4, 9 and each of these is less than 16, while (±4)² = 16 which is not less than 16.

Hence, the required set in roster form is {−3, −2, −1, 0, 1, 2, 3}.

Use the roster method to represent the set:

The set of all digits in our number system.

Answer

In our number system every number is written with the help of the ten digits 0, 1, 2, ....., 9.

Hence, the required set in roster form is {0, 1, 2, ....., 9}.

Use the roster method to represent the set:

The set of all letters in the word TRIGONOMETRY.

Answer

The letters of the word TRIGONOMETRY are T, R, I, G, O, N, O, M, E, T, R, Y. While listing the elements of a set, it is sufficient to list each member only once, so the repeated letters O, T and R are written only once.

Hence, the required set in roster form is {T, R, I, G, O, N, M, E, Y}.

Use the roster method to represent the set:

The set of all vowels in the English alphabet which precede q.

Answer

The vowels in the English alphabet are a, e, i, o, u. Out of these, a, e, i and o come before q while u comes after q.

Hence, the required set in roster form is {a, e, i, o}.

Use the roster method to represent the set:

{x : x is a consonant in the English alphabet which precedes k}.

Answer

The letters of the English alphabet which precede k are a, b, c, d, e, f, g, h, i, j. Out of these, the consonants are b, c, d, f, g, h and j.

Hence, the required set in roster form is {b, c, d, f, g, h, j}.

Write the following sets in the builder form:

(i) {1, 3, 5, 7, 9, 11, 13}

(ii) {2, 4, 6, 8, ...}

(iii) {3, 6, 9, 12, 15}

(iv) {2, 4, 8, 16, 32, 64}

(v) {5, 25, 125, 625}

(vi) {1, 4, 9, 16, ..., 100}

(vii) {1, 4, 9, 16, 25, ...}

(viii) $\bigg\lbrace \dfrac{1}{2}, \dfrac{2}{3}, \dfrac{3}{4}, \dfrac{4}{5}, ...\bigg\rbrace$.

Answer

(i) {1, 3, 5, 7, 9, 11, 13}

Each member of the given set is an odd natural number and the members begin with 1 and end with 13.

Hence, the given set in builder form is {x : x is an odd natural number less than 14}.

(ii) {2, 4, 6, 8, ...}

Each member of the given set is an even natural number and the listing does not end.

Hence, the given set in builder form is {x : x is an even natural number}.

(iii) {3, 6, 9, 12, 15}

The given members are

3 = 3 × 1, 6 = 3 × 2, 9 = 3 × 3, 12 = 3 × 4, 15 = 3 × 5, so each member is of the form 3n where n takes the values 1 to 5.

Hence, the given set in builder form is {x : x = 3n, n ∈ N and n ≤ 5}.

(iv) {2, 4, 8, 16, 32, 64}

The given members are

2 = 2¹, 4 = 2², 8 = 2³, 16 = 2⁴, 32 = 2⁵, 64 = 2⁶, so each member is of the form 2ⁿ where n takes the values 1 to 6.

Hence, the given set in builder form is {x : x = 2ⁿ, n ∈ N and n ≤ 6}.

(v) {5, 25, 125, 625}

The given members are

5 = 5¹, 25 = 5², 125 = 5³, 625 = 5⁴, so each member is of the form 5ⁿ where n takes the values 1 to 4.

Hence, the given set in builder form is {x : x = 5ⁿ, n ∈ N and n ≤ 4}.

(vi) {1, 4, 9, 16, ..., 100}

The given members are

1 = 1², 4 = 2², 9 = 3², 16 = 4², ....., 100 = 10², so each member is of the form n² where n takes the values 1 to 10.

Hence, the given set in builder form is {x : x = n², n ∈ N and n ≤ 10}.

(vii) {1, 4, 9, 16, 25, ...}

The given members are

1 = 1², 4 = 2², 9 = 3², 16 = 4², 25 = 5², ..... and the listing does not end, so each member is of the form n² where n is any natural number.

Hence, the given set in builder form is {x : x = n², n ∈ N}.

(viii) $\bigg\lbrace\dfrac{1}{2}, \dfrac{2}{3}, \dfrac{3}{4}, \dfrac{4}{5}, ...\bigg\rbrace$

In each member of the given set the denominator is one more than the numerator, and the numerator begins with 1 and goes on increasing without end. So each member is of the form $\dfrac{n}{n + 1}$ where n is any natural number.

Hence, the given set in builder form is $\bigg\lbrace x : x = \dfrac{n}{n + 1}, n \in \textbf{N}\bigg\rbrace$.

Which of the following are examples of the null set?

(i) Set of even prime numbers.

(ii) Set of odd natural numbers divisible by 2.

(iii) Set of all Indian kids 5 metres tall.

(iv) {x : x ∈ N, x < 5 and x > 8}.

(v) {x : x is a point common to any two parallel straight lines}.

(vi) {x : x is a student of your school presently studying in both classes XI and XII}.

Answer

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

(i) Set of even prime numbers:

2 is a prime number and it is even, so the given set contains the element 2 i.e. the set is {2}.

Hence, this set is not a null set.

(ii) Set of odd natural numbers divisible by 2:

Every number divisible by 2 is an even number, so no odd natural number can be divisible by 2. Therefore the given set contains no element.

Hence, this set is a null set.

(iii) Set of all Indian kids 5 metres tall:

No kid in India is 5 metres tall, so the given set contains no element.

Hence, this set is a null set.

(iv) {x : x ∈ N, x < 5 and x > 8}

A natural number x cannot satisfy x < 5 and x > 8 at the same time, since a number less than 5 can never be greater than 8. So the given set contains no element.

Hence, this set is a null set.

(v) {x : x is a point common to any two parallel straight lines}

Two parallel straight lines never intersect, so they have no point in common. Therefore the given set contains no element.

Hence, this set is a null set.

(vi) {x : x is a student of your school presently studying in both classes XI and XII}

At a given time a student can be studying in only one class, so no student can be studying in both classes XI and XII at present. Therefore the given set contains no element.

Hence, this set is a null set.

Which of the following sets are finite or infinite?

(i) The set of days of a week.

(ii) The set of numbers which are multiples of 7.

(iii) The set of animals living on Earth.

(iv) The set of consonants in the English alphabet.

(v) The set of circles drawn in a plane.

(vi) The set of prime numbers which are less than one crore.

Answer

A set that contains a limited (definite) number of different elements is called a finite set, and a set that contains an unlimited number of different elements is called an infinite set.

(i) The set of days of a week:

There are exactly 7 days in a week, namely Monday, Tuesday, Wednesday, Thursday, Friday, Saturday and Sunday. So the number of elements is limited.

Hence, the set of days of a week is a finite set.

(ii) The set of numbers which are multiples of 7:

The multiples of 7 are 7, 14, 21, 28, ..... and this listing never comes to an end.

Hence, the set of numbers which are multiples of 7 is an infinite set.

(iii) The set of animals living on Earth:

At any given moment the number of animals living on Earth, though very large, is a definite number and can be counted.

Hence, the set of animals living on Earth is a finite set.

(iv) The set of consonants in the English alphabet:

The English alphabet has 26 letters out of which 5 are vowels, so the number of consonants is 26 − 5 = 21, which is a definite number.

Hence, the set of consonants in the English alphabet is a finite set.

(v) The set of circles drawn in a plane:

In a plane a circle can be drawn with any point as centre and any positive number as radius, so an unlimited number of circles can be drawn.

Hence, the set of circles drawn in a plane is an infinite set.

(vi) The set of prime numbers which are less than one crore:

The prime numbers less than one crore are 2, 3, 5, 7, 11, ....., and since only those below one crore are taken, they are definite in number.

Hence, the set of prime numbers which are less than one crore is a finite set.

Find the cardinal number of the following sets:

(i) { }

(ii) {0}

(iii) A = {1, 2, 2, 1, 3}

(iv) The set of all Indians having 8 legs.

(v) The set of all letters in the word PRINCIPAL

(vi) The set of all vowels in the word PRINCIPAL.

Answer

The number of different elements in a finite set A is called the cardinal number of A and is denoted by n(A).

(i) { }

The set { } contains no element at all, so it is the empty set.

Hence, the cardinal number of { } is 0.

(ii) {0}

The set {0} contains exactly one element, namely the number 0.

Hence, the cardinal number of {0} is 1.

(iii) A = {1, 2, 2, 1, 3}

If one or more elements of a set are repeated, the set remains the same. So on writing each element only once,

A = {1, 2, 3}, which has 3 different elements.

Hence, the cardinal number of A is 3.

(iv) The set of all Indians having 8 legs:

No Indian has 8 legs, so this set is the empty set.

Hence, the cardinal number of the set of all Indians having 8 legs is 0.

(v) The set of all letters in the word PRINCIPAL:

The letters of the word PRINCIPAL are P, R, I, N, C, I, P, A, L. Writing each letter only once, the set is {P, R, I, N, C, A, L}, which has 7 different elements.

Hence, the cardinal number of the set of all letters in the word PRINCIPAL is 7.

(vi) The set of all vowels in the word PRINCIPAL:

The vowels occurring in the word PRINCIPAL are I, I and A. Writing each letter only once, the set is {I, A}, which has 2 different elements.

Hence, the cardinal number of the set of all vowels in the word PRINCIPAL is 2.

Write the cardinal number of the set A, where A = {x : x is a two digit number, sum of whose digits is 8}.

Answer

A two digit number has a tens digit from 1 to 9 and a units digit from 0 to 9. Taking the pairs of digits whose sum is 8, the numbers are

17, 26, 35, 44, 53, 62, 71, 80.

∴ A = {17, 26, 35, 44, 53, 62, 71, 80}, which has 8 elements.

Hence, the cardinal number of the set A is 8.

Write the cardinal number of the set of all integers x for which $\dfrac{30}{x}$ is a natural number.

Answer

$\dfrac{30}{x}$ is a natural number only when x is a positive divisor of 30.

Resolving 30 into prime factors, 30 = 2 × 3 × 5, so the positive divisors of 30 are

1, 2, 3, 5, 6, 10, 15, 30, which are 8 in number.

Hence, the cardinal number of the required set is 8.

What is the cardinal number of the set X, where X = {x : x is a letter in the word 'CHANDIGARH'}?

Answer

The letters of the word CHANDIGARH are C, H, A, N, D, I, G, A, R, H. Writing each letter only once,

X = {C, H, A, N, D, I, G, R}, which has 8 different elements.

Hence, the cardinal number of the set X is 8.

State which of the following statements are true and which are false. Justify your answer.

(i) 37 ∉ {x : x has exactly two positive factors}

(ii) 35 ∈ {x : x has exactly four positive factors}

(iii) 496 ∈ {y : the sum of all positive factors of y is 2y}.

(iv) 3 ∉ {x : x⁴ − 5x³ + 2x² − 112x + 6 = 0}

Answer

(i) 37 ∉ {x : x has exactly two positive factors}

The positive factors of 37 are 1 and 37 only, because 37 is a prime number. So 37 has exactly two positive factors and therefore 37 belongs to the given set i.e. 37 ∈ {x : x has exactly two positive factors}.

Hence, the statement 37 ∉ {x : x has exactly two positive factors} is false.

(ii) 35 ∈ {x : x has exactly four positive factors}

Since 35 = 5 × 7, the positive factors of 35 are 1, 5, 7 and 35. So 35 has exactly four positive factors and therefore 35 belongs to the given set.

Hence, the statement 35 ∈ {x : x has exactly four positive factors} is true.

(iii) 496 ∈ {y : the sum of all positive factors of y is 2y}

Resolving 496 into prime factors,

496 = 2⁴ × 31

∴ All the positive factors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496.

The sum of all these positive factors = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 + 496

= 992

= 2 × 496

So the sum of all the positive factors of 496 is twice 496, and therefore 496 belongs to the given set.

Hence, the statement 496 ∈ {y : the sum of all positive factors of y is 2y} is true.

(iv) 3 ∉ {x : x⁴ − 5x³ + 2x² − 112x + 6 = 0}

Putting x = 3 in x⁴ − 5x³ + 2x² − 112x + 6, we get:

⇒ 3⁴ − 5 × 3³ + 2 × 3² − 112 × 3 + 6

⇒ 81 − 135 + 18 − 336 + 6

⇒ −366 ≠ 0

So x = 3 does not satisfy the given equation and therefore 3 does not belong to the given set.

Hence, the statement 3 ∉ {x : x⁴ − 5x³ + 2x² − 112x + 6 = 0} is true.

Classify the following set into finite set or infinite set. In case of a finite set, mention the cardinal number.

A = {x : x ∈ I, x < 5}.

Answer

Here x is an integer and x < 5, so

A = {....., −2, −1, 0, 1, 2, 3, 4}

As the integers go on decreasing without end, the listing never comes to an end.

Hence, the set A is an infinite set.

Classify the following set into finite set or infinite set. In case of a finite set, mention the cardinal number.

A = {x : x ∈ W, x is divisible by 4 and 9}.

Answer

A whole number divisible by both 4 and 9 must be divisible by their L.C.M.

L.C.M. of 4 and 9 = 36

∴ A = {0, 36, 72, 108, .....}

As the multiples of 36 go on without end, the listing never comes to an end.

Hence, the set A is an infinite set.

Classify the following set into finite set or infinite set. In case of a finite set, mention the cardinal number.

P = {x : x is an even prime number > 2}.

Answer

2 is the only even prime number, and every even number greater than 2 is divisible by 2 and hence is not prime. So there is no even prime number greater than 2.

∴ P = φ, which is a finite set with n(P) = 0.

Hence, the set P is a finite set and its cardinal number is 0.

Classify the following set into finite set or infinite set. In case of a finite set, mention the cardinal number.

F = {x : x ∈ N and x is a factor of 84}.

Answer

Resolving 84 into prime factors,

84 = 2² × 3 × 7

∴ F = {1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84}, which has 12 elements.

Hence, the set F is a finite set and its cardinal number is 12.

Classify the following set into finite set or infinite set. In case of a finite set, mention the cardinal number.

B = {x : x is a two digit number, sum of whose digits is 12}.

Answer

Taking the two digit numbers whose digits add up to 12, we get

B = {39, 48, 57, 66, 75, 84, 93}, which has 7 elements.

Hence, the set B is a finite set and its cardinal number is 7.

Classify the following set into finite set or infinite set. In case of a finite set, mention the cardinal number.

C = {x : x ∈ W, 3x − 7 ≤ 8}.

Answer

Here x ∈ W and 3x − 7 ≤ 8

⇒ 3x ≤ 15

⇒ x ≤ 5

As x is a whole number not exceeding 5,

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

Hence, the set C is a finite set and its cardinal number is 6.

Classify the following set into finite set or infinite set. In case of a finite set, mention the cardinal number.

{x : x = 5n, n ∈ N and x < 20}.

Answer

Here x = 5n where n ∈ N and x < 20.

Putting n = 1, 2, 3, we get x = 5, 10, 15 and for n = 4 we get x = 20 which is not less than 20.

∴ The set = {5, 10, 15}, which has 3 elements.

Hence, the given set is a finite set and its cardinal number is 3.

Classify the following set into finite set or infinite set. In case of a finite set, mention the cardinal number.

{x : x = 5n, n ∈ I and x < 20}.

Answer

Here x = 5n where n is an integer and x < 20.

Since n can be any integer, taking n = 3, 2, 1, 0, −1, −2, ..... we get

the set = {....., −10, −5, 0, 5, 10, 15}

As the negative multiples of 5 go on without end, the listing never comes to an end.

Hence, the given set is an infinite set.

Classify the following set into finite set or infinite set. In case of a finite set, mention the cardinal number.

{x : x = $\dfrac{n}{n + 1}$, n ∈ W and n ≤ 10}.

Answer

Here n ∈ W and n ≤ 10, so n takes the 11 values 0, 1, 2, ....., 10.

Putting these values in $\dfrac{n}{n + 1}$, we get

$0, \dfrac{1}{2}, \dfrac{2}{3}, \dfrac{3}{4}, \dfrac{4}{5}, \dfrac{5}{6}, \dfrac{6}{7}, \dfrac{7}{8}, \dfrac{8}{9}, \dfrac{9}{10}, \dfrac{10}{11}$

All these 11 values are different.

Hence, the given set is a finite set and its cardinal number is 11.

Classify the following set into finite set or infinite set. In case of a finite set, mention the cardinal number.

{x : x = $\dfrac{2n}{n + 3}$, n ∈ N and 5 < n < 20}.

Answer

Here n ∈ N and 5 < n < 20, so n takes the 14 values 6, 7, 8, ....., 19.

Putting these values in $\dfrac{2n}{n + 3}$, we get

$\dfrac{12}{9}, \dfrac{14}{10}, \dfrac{16}{11}, \dfrac{18}{12}, \dfrac{20}{13}, \dfrac{22}{14}, \dfrac{24}{15}, \dfrac{26}{16}, \dfrac{28}{17}, \dfrac{30}{18}, \dfrac{32}{19}, \dfrac{34}{20}, \dfrac{36}{21}, \dfrac{38}{22}$

All these 14 values are different.

Hence, the given set is a finite set and its cardinal number is 14.