Sets — Exercise 5(A)

Is the following collection of objects a set?

All the months in a year.

Answer

A collection is a set only if it is well-defined, i.e., we can tell beyond doubt whether a given object belongs to the collection or not.

The months in a year are January, February, March, ..., December.

Given any object, we can definitely say whether it is a month in a year or not.

∴ The collection is well-defined.

Hence, the collection of all the months in a year is a set.

Is the following collection of objects a set?

All the rivers flowing in India.

Answer

A collection is a set only if it is well-defined, i.e., we can tell beyond doubt whether a given object belongs to the collection or not.

Given any river, we can definitely say whether it flows in India or not.

∴ The collection is well-defined.

Hence, the collection of all the rivers flowing in India is a set.

Is the following collection of objects a set?

All the planets in the solar system.

Answer

A collection is a set only if it is well-defined, i.e., we can tell beyond doubt whether a given object belongs to the collection or not.

The planets in the solar system are Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus and Neptune.

Given any object, we can definitely say whether it is a planet in the solar system or not.

∴ The collection is well-defined.

Hence, the collection of all the planets in the solar system is a set.

Is the following collection of objects a set?

All the interesting dramas written by Shakespeare.

Answer

A collection is a set only if it is well-defined, i.e., we can tell beyond doubt whether a given object belongs to the collection or not.

Here, the term 'interesting' is vague, since a drama may be interesting to one person but not to another.

So, we are not definite as to which dramas are there in the collection.

∴ The collection is not well-defined.

Hence, the collection of all the interesting dramas written by Shakespeare is not a set.

Is the following collection of objects a set?

All the short boys of your class.

Answer

A collection is a set only if it is well-defined, i.e., we can tell beyond doubt whether a given object belongs to the collection or not.

Here, the term 'short' is vague, as no definite height is given below which a boy is to be called short.

So, we cannot judge definitely as to which boys of the class are there in the collection.

∴ The collection is not well-defined.

Hence, the collection of all the short boys of your class is not a set.

Is the following collection of objects a set?

All the letters of English Alphabet which precede K.

Answer

A collection is a set only if it is well-defined, i.e., we can tell beyond doubt whether a given object belongs to the collection or not.

The letters of the English alphabet which precede K are A, B, C, D, E, F, G, H, I and J.

Given any letter, we can definitely say whether it precedes K or not.

∴ The collection is well-defined.

Hence, the collection of all the letters of English Alphabet which precede K is a set.

Is the following collection of objects a set?

All the pet dogs in Meerut.

Answer

A collection is a set only if it is well-defined, i.e., we can tell beyond doubt whether a given object belongs to the collection or not.

Given any dog, we can definitely say whether it is a pet dog in Meerut or not.

∴ The collection is well-defined.

Hence, the collection of all the pet dogs in Meerut is a set.

Is the following collection of objects a set?

All the dishonest dealers in Delhi.

Answer

A collection is a set only if it is well-defined, i.e., we can tell beyond doubt whether a given object belongs to the collection or not.

Here, the term 'dishonest' is vague, since a dealer may be considered dishonest by one person but not by another.

So, we are not definite as to which dealers are there in the collection.

∴ The collection is not well-defined.

Hence, the collection of all the dishonest dealers in Delhi is not a set.

Is the following collection of objects a set?

All the students of your school with age exceeding 12 years.

Answer

A collection is a set only if it is well-defined, i.e., we can tell beyond doubt whether a given object belongs to the collection or not.

Given any student of the school, we can definitely say whether his or her age exceeds 12 years or not.

∴ The collection is well-defined.

Hence, the collection of all the students of your school with age exceeding 12 years is a set.

Is the following collection of objects a set?

All the girls of Meena's class, who are taller than Meena.

Answer

A collection is a set only if it is well-defined, i.e., we can tell beyond doubt whether a given object belongs to the collection or not.

The height of Meena is a definite height and the height of every girl of her class can be compared with it.

So, given any girl of Meena's class, we can definitely say whether she is taller than Meena or not.

∴ The collection is well-defined.

Hence, the collection of all the girls of Meena's class, who are taller than Meena, is a set.

Rewrite the following statement using set notation:

p is an element of set A.

Answer

If x is an element of a set A, we say that x belongs to A and we write x ∈ A.

Hence, p is an element of set A is written as p ∈ A.

Rewrite the following statement using set notation:

q does not belong to set B.

Answer

If x is not an element of a set B, we say that x does not belong to B and we write x ∉ B.

Hence, q does not belong to set B is written as q ∉ B.

Rewrite the following statement using set notation:

a and b are members of set C.

Answer

If x is a member of a set C, we write x ∈ C.

As both a and b are members of set C, we write a ∈ C and b ∈ C.

Hence, a and b are members of set C is written as a, b ∈ C.

Rewrite the following statement using set notation:

B and C are equivalent sets.

Answer

Two finite sets B and C are said to be equivalent, if n(B) = n(C), and it is written as B ↔ C.

Hence, B and C are equivalent sets is written as B ↔ C.

Rewrite the following statement using set notation:

Cardinal number of set E is 15.

Answer

The number of distinct elements contained in a finite set E is called the cardinal number of E and is denoted by n(E).

Hence, cardinal number of set E is 15 is written as n(E) = 15.

Rewrite the following statement using set notation:

A is an empty set and B is a non-empty set.

Answer

A set consisting of no element at all is called an empty set and it is denoted by φ.

A set which has at least one element is called a non-empty set.

Hence, A is an empty set and B is a non-empty set is written as A = φ and B ≠ φ.

Rewrite the following statement using set notation:

0 is a whole number, but 0 is not a natural number

Answer

The set of all whole numbers is denoted by W and the set of all natural numbers is denoted by N.

As 0 belongs to W, we write 0 ∈ W and as 0 does not belong to N, we write 0 ∉ N.

Hence, 0 is a whole number, but 0 is not a natural number is written as 0 ∈ W but 0 ∉ N.

Describe the following set in roster form:

B = {x | x ∈ W, x ≤ 6}.

Answer

B = {x | x ∈ W, x ≤ 6}

In roster form, we make a list of the elements of the set and put it within braces { }.

The whole numbers which are less than or equal to 6 are 0, 1, 2, 3, 4, 5 and 6.

Hence, B = {0, 1, 2, 3, 4, 5, 6}.

Describe the following set in roster form:

C = {x | x is a factor of 32}.

Answer

C = {x | x is a factor of 32}

In roster form, we make a list of the elements of the set and put it within braces { }.

32 = 1 × 32 = 2 × 16 = 4 × 8

So, the factors of 32 are 1, 2, 4, 8, 16 and 32.

Hence, C = {1, 2, 4, 8, 16, 32}.

Describe the following set in roster form:

E = {x | x = (2n + 1), n ∈ W, n ≤ 4}.

Answer

E = {x | x = (2n + 1), n ∈ W, n ≤ 4}

In roster form, we make a list of the elements of the set and put it within braces { }.

The whole numbers which are less than or equal to 4 are 0, 1, 2, 3 and 4.

Putting n = 0, 1, 2, 3 and 4 in (2n + 1), we get

2 × 0 + 1 = 1, 2 × 1 + 1 = 3, 2 × 2 + 1 = 5, 2 × 3 + 1 = 7 and 2 × 4 + 1 = 9.

Hence, E = {1, 3, 5, 7, 9}.

Describe the following set in roster form:

F = {x | x = n², n ∈ N, 2 ≤ n ≤ 5}.

Answer

F = {x | x = n², n ∈ N, 2 ≤ n ≤ 5}

In roster form, we make a list of the elements of the set and put it within braces { }.

The natural numbers from 2 to 5 are 2, 3, 4 and 5.

Putting n = 2, 3, 4 and 5 in n², we get

2² = 4, 3² = 9, 4² = 16 and 5² = 25.

Hence, F = {4, 9, 16, 25}.

Describe the following set in roster form:

G = $\bigg\lbrace x\ |\ x = \dfrac{n}{(n + 3)},\ n \in N \text{ and } n \le 5\bigg\rbrace$.

Answer

G = $\bigg\lbrace x\ |\ x = \dfrac{n}{(n + 3)},\ n \in N \text{ and } n \le 5\bigg\rbrace$

In roster form, we make a list of the elements of the set and put it within braces { }.

The natural numbers which are less than or equal to 5 are 1, 2, 3, 4 and 5.

Putting n = 1, 2, 3, 4 and 5 in $\dfrac{n}{(n + 3)}$, we get

$\dfrac{1}{1 + 3} = \dfrac{1}{4}, \dfrac{2}{2 + 3} = \dfrac{2}{5}, \dfrac{3}{3 + 3} = \dfrac{3}{6}, \dfrac{4}{4 + 3} = \dfrac{4}{7} \text{ and } \dfrac{5}{5 + 3} = \dfrac{5}{8}$.

Hence, G = $\mathbf{\bigg\lbrace\dfrac{1}{4},\ \dfrac{2}{5},\ \dfrac{3}{6},\ \dfrac{4}{7},\ \dfrac{5}{8}\bigg\rbrace}$.

Describe the following set in roster form:

H = {x | x is a two-digit number, the sum of whose digits is 8}.

Answer

H = {x | x is a two-digit number, the sum of whose digits is 8}

In roster form, we make a list of the elements of the set and put it within braces { }.

A two-digit number is of the form (10a + b), where a is a whole number from 1 to 9 and b is a whole number from 0 to 9.

Here, a + b = 8. So, the possible pairs of a and b are

a = 1, b = 7; a = 2, b = 6; a = 3, b = 5; a = 4, b = 4; a = 5, b = 3; a = 6, b = 2; a = 7, b = 1 and a = 8, b = 0.

So, the required two-digit numbers are 17, 26, 35, 44, 53, 62, 71 and 80.

Hence, H = {17, 26, 35, 44, 53, 62, 71, 80}.

Describe the following set in roster form:

I = {x | x ∈ N, x is divisible both by 4 and 6 and x ≤ 60}.

Answer

I = {x | x ∈ N, x is divisible both by 4 and 6 and x ≤ 60}

In roster form, we make a list of the elements of the set and put it within braces { }.

A number divisible both by 4 and 6 is divisible by their L.C.M.

L.C.M. of 4 and 6 = 12

So, the multiples of 12 which are less than or equal to 60 are 12, 24, 36, 48 and 60.

Hence, I = {12, 24, 36, 48, 60}.

Describe the following set in roster form:

J = $\lbrace x\ |\ x = \dfrac{1}{n},\ n \in N \text{ and } n \le 5\rbrace$.

Answer

J = $\lbrace x\ |\ x = \dfrac{1}{n},\ n \in N \text{ and } n \le 5\rbrace$

In roster form, we make a list of the elements of the set and put it within braces { }.

The natural numbers which are less than or equal to 5 are 1, 2, 3, 4 and 5.

Putting n = 1, 2, 3, 4 and 5 in $\dfrac{1}{n}$, we get

$\dfrac{1}{1} = 1, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4} \text{ and } \dfrac{1}{5}$.

Hence, J = $\mathbf{\bigg\lbrace1,\ \dfrac{1}{2},\ \dfrac{1}{3},\ \dfrac{1}{4},\ \dfrac{1}{5}\bigg\rbrace}$.

Describe the following set in roster form:

L = {x | x is a letter of the word 'careless'}.

Answer

L = {x | x is a letter of the word 'careless'}

In roster form, we make a list of the elements of the set and put it within braces { }.

The letters of the word 'careless' are c, a, r, e, l, e, s, s.

As every repeated element is taken only once in a set, writing each letter once and only once, we get c, a, r, e, l and s.

Hence, L = {c, a, r, e, l, s}.

Describe the following set in set-builder form:

A = {5, 6, 7, 8, 9, 10, 11, 12}.

Answer

A = {5, 6, 7, 8, 9, 10, 11, 12}

In set-builder form, we list the property satisfied by all the elements of the set and put it within braces.

Each element of A is a natural number lying between 4 and 13.

Hence, A = {x | x ∈ N, 4 < x < 13}.

Describe the following set in set-builder form:

B = {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}.

Answer

B = {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}

In set-builder form, we list the property satisfied by all the elements of the set and put it within braces.

48 = 1 × 48 = 2 × 24 = 3 × 16 = 4 × 12 = 6 × 8

So, the elements of B are exactly the factors of 48.

Hence, B = {x | x ∈ N, x is a factor of 48}.

Describe the following set in set-builder form:

C = {11, 13, 17, 19, 23, 29, 31, 37}.

Answer

C = {11, 13, 17, 19, 23, 29, 31, 37}

In set-builder form, we list the property satisfied by all the elements of the set and put it within braces.

Each element of C is a prime number and these are all the prime numbers lying between 10 and 40.

Hence, C = {x | x is a prime number, 10 < x < 40}.

Describe the following set in set-builder form:

D = {21, 23, 25, 27, 29, 31, 33, 35, 37}.

Answer

D = {21, 23, 25, 27, 29, 31, 33, 35, 37}

In set-builder form, we list the property satisfied by all the elements of the set and put it within braces.

Each element of D is an odd number and these are all the odd numbers lying between 20 and 38.

Hence, D = {x | x is an odd number, 20 < x < 38}.

Describe the following set in set-builder form:

I = {9, 16, 25, 36, 49, 64, 81, 100}.

Answer

I = {9, 16, 25, 36, 49, 64, 81, 100}

In set-builder form, we list the property satisfied by all the elements of the set and put it within braces.

9 = 3², 16 = 4², 25 = 5², 36 = 6², 49 = 7², 64 = 8², 81 = 9² and 100 = 10²

So, each element of I is the square of a natural number n, where n takes the values from 3 to 10.

Hence, I = {x | x = n², n ∈ N, 3 ≤ n ≤ 10}.

Describe the following set in set-builder form:

J = {-2, 2}.

Answer

J = {-2, 2}

In set-builder form, we list the property satisfied by all the elements of the set and put it within braces.

(-2)² = 4 and (2)² = 4

So, each element of J is an integer whose square is 4.

Hence, J = {x | x ∈ Z, x² = 4}.

Describe the following set in set-builder form:

K = {0}.

Answer

K = {0}

In set-builder form, we list the property satisfied by all the elements of the set and put it within braces.

The only element of K is 0, which is the only whole number less than 1.

Hence, K = {x | x ∈ W, x < 1}.

Describe the following set in set-builder form:

L = { }.

Answer

L = { }

In set-builder form, we list the property satisfied by all the elements of the set and put it within braces.

L has no element at all, so we describe it by a property which no object satisfies.

As there is no natural number which is less than 1,

Hence, L = {x | x ∈ N, x < 1}.

Describe the following set in set-builder form:

M = {a, b, c, d, e, f, g, h, i}.

Answer

M = {a, b, c, d, e, f, g, h, i}

In set-builder form, we list the property satisfied by all the elements of the set and put it within braces.

The elements of M are the first nine letters of the English alphabet.

Hence, M = {x | x is one of the first nine letters of the English alphabet}.

Describe the following set in set-builder form:

P = $\bigg\lbrace \dfrac{2}{7},\ \dfrac{3}{8},\ \dfrac{4}{9},\ \dfrac{5}{10},\ \dfrac{6}{11},\ \dfrac{7}{12},\ \dfrac{8}{13},\ \dfrac{9}{14} \bigg\rbrace$.

Answer

P = $\bigg\lbrace \dfrac{2}{7},\ \dfrac{3}{8},\ \dfrac{4}{9},\ \dfrac{5}{10},\ \dfrac{6}{11},\ \dfrac{7}{12},\ \dfrac{8}{13},\ \dfrac{9}{14} \bigg\rbrace$

In set-builder form, we list the property satisfied by all the elements of the set and put it within braces.

The numerators are 2, 3, 4, ..., 9 and each denominator is 5 more than its numerator.

Taking the numerator as (n + 1), the denominator becomes (n + 6), where n = 1, 2, 3, ..., 8.

Hence, P = $\mathbf{\bigg\lbrace x\ |\ x = \dfrac{n + 1}{n + 6},\ n \in N \text{ and } n \le 8 \bigg\rbrace}$.

Describe the following set in set-builder form:

S = {Atlantic, Arctic}.

Answer

S = {Atlantic, Arctic}

In set-builder form, we list the property satisfied by all the elements of the set and put it within braces.

The oceans of the world are Pacific, Atlantic, Indian, Southern and Arctic, out of which only Atlantic and Arctic begin with the letter A.

Hence, S = {x | x is an ocean whose name begins with the letter A}.

Describe the following set in set-builder form:

T = {Mars, Mercury}.

Answer

T = {Mars, Mercury}

In set-builder form, we list the property satisfied by all the elements of the set and put it within braces.

The planets of the solar system are Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus and Neptune, out of which only Mercury and Mars begin with the letter M.

Hence, T = {x | x is a planet of the solar system whose name begins with the letter M}.

Is the following set finite or infinite?

Set of leaves on a tree.

Answer

A set in which the process of counting of elements surely comes to an end, is called a finite set.

A set which consists of an uncountable number of distinct elements, is called an infinite set.

Although the leaves on a tree are very large in number, the process of counting them will surely come to an end.

Hence, the set of leaves on a tree is a finite set.

Is the following set finite or infinite?

Set of all counting numbers.

Answer

A set in which the process of counting of elements surely comes to an end, is called a finite set.

A set which consists of an uncountable number of distinct elements, is called an infinite set.

The set of all counting numbers is N = {1, 2, 3, 4, 5, ....} and the process of counting its elements never comes to an end.

Hence, the set of all counting numbers is an infinite set.

Is the following set finite or infinite?

{x | x ∈ N, x > 1000}.

Answer

A set in which the process of counting of elements surely comes to an end, is called a finite set.

A set which consists of an uncountable number of distinct elements, is called an infinite set.

{x | x ∈ N, x > 1000} = {1001, 1002, 1003, ....} and the process of counting its elements never comes to an end.

Hence, {x | x ∈ N, x > 1000} is an infinite set.

Is the following set finite or infinite?

{x | x ∈ W, x < 5000}.

Answer

A set in which the process of counting of elements surely comes to an end, is called a finite set.

A set which consists of an uncountable number of distinct elements, is called an infinite set.

{x | x ∈ W, x < 5000} = {0, 1, 2, 3, ...., 4999}, which has 5000 elements. So, the process of counting its elements comes to an end.

Hence, {x | x ∈ W, x < 5000} is a finite set.

Is the following set finite or infinite?

{x | x ∈ Z, x < 4}.

Answer

A set in which the process of counting of elements surely comes to an end, is called a finite set.

A set which consists of an uncountable number of distinct elements, is called an infinite set.

{x | x ∈ Z, x < 4} = {...., -3, -2, -1, 0, 1, 2, 3} and the process of counting its elements never comes to an end.

Hence, {x | x ∈ Z, x < 4} is an infinite set.

Is the following set finite or infinite?

Set of all triangles in a plane.

Answer

A set in which the process of counting of elements surely comes to an end, is called a finite set.

A set which consists of an uncountable number of distinct elements, is called an infinite set.

An unlimited number of triangles can be drawn in a plane, so the process of counting them never comes to an end.

Hence, the set of all triangles in a plane is an infinite set.

Is the following set finite or infinite?

Set of all points on the circumference of a circle.

Answer

A set in which the process of counting of elements surely comes to an end, is called a finite set.

A set which consists of an uncountable number of distinct elements, is called an infinite set.

There is an unlimited number of points on the circumference of a circle, so the process of counting them never comes to an end.

Hence, the set of all points on the circumference of a circle is an infinite set.

Is the following set finite or infinite?

{1, 2, 3, 1, 2, 3, 1, 2, 3, ....}.

Answer

A set in which the process of counting of elements surely comes to an end, is called a finite set.

A set which consists of an uncountable number of distinct elements, is called an infinite set.

As every repeated element is taken only once in a set, {1, 2, 3, 1, 2, 3, 1, 2, 3, ....} = {1, 2, 3}, which has only 3 elements.

Hence, {1, 2, 3, 1, 2, 3, 1, 2, 3, ....} is a finite set.

Is the following set finite or infinite?

{x | x ∈ Q, 2 < x < 3}.

Answer

A set in which the process of counting of elements surely comes to an end, is called a finite set.

A set which consists of an uncountable number of distinct elements, is called an infinite set.

Between any two rational numbers there lies an unlimited number of rational numbers. So, the process of counting the rational numbers lying between 2 and 3 never comes to an end.

Hence, {x | x ∈ Q, 2 < x < 3} is an infinite set.

Is the following an empty set?

A = {x | x ∈ N, x + 5 = 5}

Answer

A set consisting of no element at all is called an empty set and it is denoted by φ.

A = {x | x ∈ N, x + 5 = 5}

⇒ x + 5 = 5

⇒ x = 5 - 5

⇒ x = 0

As 0 is not a natural number, there is no natural number x satisfying x + 5 = 5.

Hence, A = {x | x ∈ N, x + 5 = 5} is an empty set.

Is the following an empty set?

B = {x | x ∈ N, 2x + 3 = 6}

Answer

A set consisting of no element at all is called an empty set and it is denoted by φ.

B = {x | x ∈ N, 2x + 3 = 6}

2x + 3 = 6 ⇒ 2x = 6 - 3 = 3 ⇒ x = $\dfrac{3}{2}$

As $\dfrac{3}{2}$ is not a natural number, there is no natural number x satisfying 2x + 3 = 6.

Hence, B = {x | x ∈ N, 2x + 3 = 6} is an empty set.

Is the following an empty set?

C = {x | x ∈ W, x + 2 < 2}.

Answer

A set consisting of no element at all is called an empty set and it is denoted by φ.

C = {x | x ∈ W, x + 2 < 2}

x + 2 < 2 ⇒ x < 2 - 2 ⇒ x < 0

As there is no whole number which is less than 0, there is no whole number x satisfying x + 2 < 2.

Hence, C = {x | x ∈ W, x + 2 < 2} is an empty set.

Is the following an empty set?

D = {x | x ∈ N, 1 < x ≤ 2}.

Answer

A set consisting of no element at all is called an empty set and it is denoted by φ.

D = {x | x ∈ N, 1 < x ≤ 2}

The natural number which is greater than 1 and less than or equal to 2 is 2.

∴ D = {2}

As D has one element, D is not an empty set.

Hence, D = {x | x ∈ N, 1 < x ≤ 2} is not an empty set.

Is the following an empty set?

E = {x | x ∈ N, x² + 4 = 0}.

Answer

A set consisting of no element at all is called an empty set and it is denoted by φ.

E = {x | x ∈ N, x² + 4 = 0}

x² + 4 = 0 ⇒ x² = -4

As the square of a natural number is always positive, there is no natural number x satisfying x² + 4 = 0.

Hence, E = {x | x ∈ N, x² + 4 = 0} is an empty set.

Is the following an empty set?

F = {x | x is a prime number, 90 < x < 96}.

Answer

A set consisting of no element at all is called an empty set and it is denoted by φ.

F = {x | x is a prime number, 90 < x < 96}

The numbers lying between 90 and 96 are 91, 92, 93, 94 and 95.

91 = 7 × 13, 92 = 2 × 46, 93 = 3 × 31, 94 = 2 × 47 and 95 = 5 × 19

So, none of these numbers is a prime number.

Hence, F = {x | x is a prime number, 90 < x < 96} is an empty set.

Is the following an empty set?

G = {x | x is an even prime}.

Answer

A set consisting of no element at all is called an empty set and it is denoted by φ.

G = {x | x is an even prime}

2 is a prime number and it is even.

∴ G = {2}

As G has one element, G is not an empty set.

Hence, G = {x | x is an even prime} is not an empty set.

Is the following an empty set?

H = {0}.

Answer

A set consisting of no element at all is called an empty set and it is denoted by φ.

H = {0}

H is a set containing one element, namely 0. So, H is not a set consisting of no element at all.

Hence, H = {0} is not an empty set.

Is the following a pair of equivalent sets?

A = {2, 3, 5, 7} and B = {x : x is a whole number, x ≤ 4}.

Answer

Two finite sets A and B are said to be equivalent, if n(A) = n(B), i.e., they have the same number of elements.

A = {2, 3, 5, 7}

As A has 4 elements, n(A) = 4.

B = {x : x is a whole number, x ≤ 4}

B = {0, 1, 2, 3, 4}

As B has 5 elements, n(B) = 5.

As n(A) ≠ n(B),

Hence, A and B are not equivalent sets.

Is the following a pair of equivalent sets?

C = {x : x + 2 = 2} and D = φ.

Answer

Two finite sets C and D are said to be equivalent, if n(C) = n(D), i.e., they have the same number of elements.

C = {x : x + 2 = 2}

x + 2 = 2 ⇒ x = 2 - 2 = 0

∴ C = {0}

As C has 1 element, n(C) = 1.

D = φ

As D has no element, n(D) = 0.

As n(C) ≠ n(D),

Hence, C and D are not equivalent sets.

Is the following a pair of equivalent sets?

E = {x : x is a natural number, x < 4} and F = {x : x is a whole number, x < 3}.

Answer

Two finite sets E and F are said to be equivalent, if n(E) = n(F), i.e., they have the same number of elements.

E = {x : x is a natural number, x < 4}

E = {1, 2, 3}

As E has 3 elements, n(E) = 3.

F = {x : x is a whole number, x < 3}

F = {0, 1, 2}

As F has 3 elements, n(F) = 3.

As n(E) = n(F),

Hence, E and F are equivalent sets.

Is the following a pair of equivalent sets?

G = {x : x is an integer, -3 < x < 3} and H = {x : x is a factor of 16}.

Answer

Two finite sets G and H are said to be equivalent, if n(G) = n(H), i.e., they have the same number of elements.

G = {x : x is an integer, -3 < x < 3}

G = {-2, -1, 0, 1, 2}

As G has 5 elements, n(G) = 5.

H = {x : x is a factor of 16}

16 = 1 × 16 = 2 × 8 = 4 × 4

H = {1, 2, 4, 8, 16}

As H has 5 elements, n(H) = 5.

As n(G) = n(H),

Hence, G and H are equivalent sets.

State whether the following statement is true or false:

{a, b, c, 1, 2, 3} is not a set.

Answer

A set is a well-defined collection of distinct objects.

{a, b, c, 1, 2, 3} is a collection of six distinct objects and given any object, we can definitely say whether it belongs to this collection or not.

So, {a, b, c, 1, 2, 3} is a set.

Hence, the given statement is false.

State whether the following statement is true or false:

{5, 7, 9} = {9, 5, 7}.

Answer

The order in which the elements of a set are listed is immaterial.

Each of 5, 7 and 9 belongs to {5, 7, 9} as well as to {9, 5, 7}.

So, every element of {5, 7, 9} is in {9, 5, 7} and every element of {9, 5, 7} is in {5, 7, 9}.

Hence, the given statement is true.

State whether the following statement is true or false:

{x | x ∈ W, x + 8 = 8} is a singleton set.

Answer

A set containing exactly one element is called a singleton set.

⇒ x + 8 = 8

⇒ x = 8 - 8

⇒ x = 0

As 0 is a whole number,

{x | x ∈ W, x + 8 = 8} = {0}, which contains exactly one element.

Hence, the given statement is true.

State whether the following statement is true or false:

{x | x ∈ W, x < 0} = φ.

Answer

A set consisting of no element at all is called an empty set and it is denoted by φ.

The whole numbers are 0, 1, 2, 3, .... and none of them is less than 0.

So, {x | x ∈ W, x < 0} has no element at all.

Hence, the given statement is true.

State whether the following statement is true or false:

{x | x ∈ N, x + 5 = 3} = φ.

Answer

A set consisting of no element at all is called an empty set and it is denoted by φ.

x + 5 = 3 ⇒ x = 3 - 5 = -2

As -2 is not a natural number, there is no natural number x satisfying x + 5 = 3.

So, {x | x ∈ N, x + 5 = 3} has no element at all.

Hence, the given statement is true.

State whether the following statement is true or false:

{x | x ∈ N, 3 < x ≤ 4} = φ.

Answer

A set consisting of no element at all is called an empty set and it is denoted by φ.

The natural number which is greater than 3 and less than or equal to 4 is 4.

So, {x | x ∈ N, 3 < x ≤ 4} = {4}, which has one element.

∴ {x | x ∈ N, 3 < x ≤ 4} ≠ φ

Hence, the given statement is false.

State whether the following statement is true or false:

If A = {x | x is a letter of the word, 'MEERUT'}, then n(A) = 6.

Answer

The repetition of elements in a set is meaningless and so while finding the cardinal number of a set we do not consider the elements that are repeated.

The letters of the word 'MEERUT' are M, E, E, R, U, T.

Writing each letter once and only once, we get

A = {M, E, R, U, T}

As A has 5 elements, n(A) = 5 and not 6.

Hence, the given statement is false.

State whether the following statement is true or false:

If A = {x | x ∈ N, 8 < x < 13} and B = {x | x ∈ Z, -3 ≤ x < 1}, then n(A) = n(B).

Answer

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

A = {x | x ∈ N, 8 < x < 13}

A = {9, 10, 11, 12}

As A has 4 elements, n(A) = 4.

B = {x | x ∈ Z, -3 ≤ x < 1}

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

As B has 4 elements, n(B) = 4.

∴ n(A) = n(B)

Hence, the given statement is true.

State whether the following statement is true or false:

If n(A) = n(B), then A = B.

Answer

Two sets A and B are said to be equal, if every element of A is in B and every element of B is in A.

If n(A) = n(B), then A and B are only equivalent sets and two equivalent sets need not be equal.

For example, let A = {1, 2, 3} and B = {a, b, c}.

Here n(A) = n(B) = 3, but 1 ∈ A and 1 ∉ B, so A ≠ B.

Hence, the given statement is false.

State whether the following statement is true or false:

If B is the set of all consonants in English alphabet, then n(B) = 21.

Answer

The number of distinct elements contained in a finite set B is called the cardinal number of B and is denoted by n(B).

There are 26 letters in the English alphabet, out of which a, e, i, o and u are the 5 vowels and the remaining letters are consonants.

∴ Number of consonants = 26 - 5 = 21

So, n(B) = 21.

Hence, the given statement is true.

State whether the following statement is true or false:

{x | x is a prime factor of 24} = {2, 3, 4, 6, 8, 12, 24}.

Answer

A prime factor of a number is a factor of that number which is a prime number.

24 = 2 × 2 × 2 × 3

So, the prime factors of 24 are 2 and 3.

∴ {x | x is a prime factor of 24} = {2, 3} ≠ {2, 3, 4, 6, 8, 12, 24}

Hence, the given statement is false.