Sets — Exercise 5(D)

If A = {1}, which of the following statements is correct:

1. A = 1

2. 1 ⊂ A

3. {1} ∈ A

4. 1 ∈ A

Answer

A = {1}

A is a set containing one element, namely 1, whereas 1 is a number and not a set.

∴ A ≠ 1, so option 1 is not correct.

The symbol ⊂ is used between two sets and 1 is not a set.

∴ 1 ⊄ A, so option 2 is not correct.

The only element of A is 1 and {1} is not an element of A.

∴ {1} ∉ A, so option 3 is not correct.

As 1 is an element of A,

∴ 1 ∈ A

Hence, option 4 is the correct option.

If A = {1, 2, {3, 4}}, which of the following is a correct statement?

1. 3 ∈ A

2. {1} ⊂ A

3. {2} ∈ A

4. 1 ⊆ A

Answer

A = {1, 2, {3, 4}}

The elements of A are 1, 2 and {3, 4}.

As 3 is not an element of A, 3 ∉ A, so option 1 is not correct.

The only element of {1} is 1 and 1 ∈ A. Also, 2 ∈ A but 2 ∉ {1}.

∴ {1} ⊂ A

As {2} is not an element of A, {2} ∉ A, so option 3 is not correct.

The symbol ⊆ is used between two sets and 1 is not a set, so option 4 is not correct.

Hence, option 2 is the correct option.

Consider the following statements:

(i) Any set A is comparable with itself.

(ii) {0} is a singleton set.

(iii) {φ} is an empty set.

Of these statements, the correct ones are:

1. (i) & (ii)

2. (i) & (iii)

3. (ii) & (iii)

4. (i), (ii) & (iii)

Answer

Every set is a subset of itself, i.e., A ⊆ A, so any set A is comparable with itself.

∴ Statement (i) is correct.

A set containing exactly one element is called a singleton set and {0} contains exactly one element, namely 0.

∴ Statement (ii) is correct.

{φ} is a set containing one element, namely φ, so it is not a set consisting of no element at all.

∴ Statement (iii) is not correct.

So, the correct statements are (i) and (ii).

Hence, option 1 is the correct option.

Total number of elements in the power set of a set A containing n elements is:

1. n²

2. 2ⁿ

3. 2ⁿ - 1

4. None of these

Answer

The set of all possible subsets of a set A is called the power set of A, denoted by P(A).

If A contains n elements, then A has 2ⁿ subsets and each of these subsets is an element of P(A).

∴ Total number of elements in P(A) = 2ⁿ

Hence, option 2 is the correct option.

If a finite set S contains n elements, then the number of non-empty proper subsets of S is

1. 2.2^n-1

2. 2(2ⁿ - 1)

3. (2^n-1 - 1)

4. 2(2^n-1 - 1)

Answer

If a set S contains n elements, then the total number of subsets of S is 2ⁿ.

Out of these subsets, the empty set φ and the set S itself are not non-empty proper subsets of S.

∴ Number of non-empty proper subsets of S = 2ⁿ - 2

= 2 × 2^n-1 - 2 × 1

= 2(2^n-1 - 1)

Hence, option 4 is the correct option.

The number of all possible proper subsets of {2, 3, 5} are

1. 3

2. 6

3. 7

4. 8

Answer

A set containing n elements has (2ⁿ - 1) proper subsets.

{2, 3, 5} contains 3 elements.

∴ Number of proper subsets = 2³ - 1 = 8 - 1 = 7

Hence, option 3 is the correct option.

If A = {a, b}, then the power set of A is

1. {a^b, b^a}

2. {a², b²}

3. {φ, {a}, {b}}

4. {φ, {a}, {b}, {a, b}}

Answer

A = {a, b}

As A contains 2 elements, there are 2² = 4 subsets of A.

All possible subsets of A are φ, {a}, {b} and {a, b}.

∴ P(A) = Set of all possible subsets of A = {φ, {a}, {b}, {a, b}}

Hence, option 4 is the correct option.

Which one of the following is a correct statement?

1. φ = 0

2. φ = {0}

3. φ = {φ}

4. φ = { }

Answer

φ is a set consisting of no element at all.

0 is a number and not a set, so φ ≠ 0 and option 1 is not correct.

{0} is a set containing one element, namely 0, so φ ≠ {0} and option 2 is not correct.

{φ} is a set containing one element, namely φ, so φ ≠ {φ} and option 3 is not correct.

In roster form, an empty set is written as { }.

∴ φ = { }

Hence, option 4 is the correct option.

Which one of the following is a correct statement?

1. {a} ∈ {a, b, c}

2. a ⊆ {a, b, c}

3. a ∈ {{a}, b}

4. None of these

Answer

The elements of {a, b, c} are a, b and c.

As {a} is not an element of {a, b, c}, {a} ∉ {a, b, c}, so option 1 is not correct.

The symbol ⊆ is used between two sets and a is not a set, so option 2 is not correct.

The elements of {{a}, b} are {a} and b.

As a is not an element of {{a}, b}, a ∉ {{a}, b}, so option 3 is not correct.

So, none of the given statements is correct.

Hence, option 4 is the correct option.

Which one of the following is a correct statement?

1. Every subset of an infinite set is infinite.

2. Every set has a proper subset.

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

4. Every subset of a finite set is finite.

Answer

{1, 2, 3} is a subset of the infinite set N = {1, 2, 3, 4, ....} and it is finite, so option 1 is not correct.

The empty set φ has no element at all, so φ has no proper subset and option 2 is not correct.

As every repeated element is taken only once in a set,

{a, b, c, 1, 2, 3, a, b, c, 1, 2, 3, ....} = {a, b, c, 1, 2, 3}, which is a finite set, so option 3 is not correct.

A subset of a finite set cannot have more elements than the finite set itself, so it is finite.

Hence, option 4 is the correct option.

Which of the following is a singleton set?

1. {x ∈ R : x² = x}

2. {x ∈ N : 3x = 4}

3. {x ∈ R : x² = -1}

4. {x : x is an integer which is neither positive nor negative}

Answer

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

x² = x ⇒ x² - x = 0 ⇒ x(x - 1) = 0 ⇒ x = 0 or x = 1

So, {x ∈ R : x² = x} = {0, 1}, which has two elements and option 1 is not correct.

3x = 4 ⇒ x = $\dfrac{4}{3}$, which is not a natural number.

So, {x ∈ N : 3x = 4} = φ and option 2 is not correct.

As the square of a real number is never negative, {x ∈ R : x² = -1} = φ and option 3 is not correct.

The only integer which is neither positive nor negative is 0.

So, {x : x is an integer which is neither positive nor negative} = {0}, which contains exactly one element.

Hence, option 4 is the correct option.

Which one of the following is an infinite set?

1. {x : x is a natural number, x < 50}

2. {x : x is an integer, x < 50}

3. {x : x is an integer, x is a factor of 500}

4. {x : x is a whole number, x < 1000}

Answer

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

{x : x is a natural number, x < 50} = {1, 2, 3, ...., 49}, which is a finite set.

{x : x is an integer, x < 50} = {...., -3, -2, -1, 0, 1, ...., 49} and the process of counting its elements never comes to an end.

The factors of 500 are limited in number, so {x : x is an integer, x is a factor of 500} is a finite set.

{x : x is a whole number, x < 1000} = {0, 1, 2, ...., 999}, which is a finite set.

Hence, option 2 is the correct option.

Which one of the following is a finite set?

1. {x : x is an integer, x < 1}

2. {x : x is a rational number, 0 < x < 1}

3. {x : x is a natural number, x > 5}

4. {x : x is an even prime}

Answer

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

{x : x is an integer, x < 1} = {...., -3, -2, -1, 0}, which is an infinite set.

Between 0 and 1 there lies an unlimited number of rational numbers, so {x : x is a rational number, 0 < x < 1} is an infinite set.

{x : x is a natural number, x > 5} = {6, 7, 8, 9, ....}, which is an infinite set.

2 is the only even prime number, so {x : x is an even prime} = {2}, which has only 1 element.

Hence, option 4 is the correct option.

If A = {1, 2, 3, 4} and B = {5, 6, 7}, then A ∩ B = ?

1. {1, 2, 3}

2. {5, 6, 7}

3. {4}

4. φ

Answer

A = {1, 2, 3, 4}

B = {5, 6, 7}

A ∩ B - contains all the common elements in set A and B.

A ∩ B = {1, 2, 3, 4} ∩ {5, 6, 7}

As there is no element common to A and B,

A ∩ B = φ

Hence, option 4 is the correct option.

Let A = {0, 1, 3, 4}, B = {5, 6, 1, 3, 9} and C = {0, 1, 2, 3, 9, 13}. Then, (A ∩ B) ∪ C is

1. {0, 1, 2, 3, 9, 13}

2. {0, 1, 3}

3. {0, 1, 2, 3, 4, 9, 13}

4. {1, 3}

Answer

A = {0, 1, 3, 4}

B = {5, 6, 1, 3, 9}

C = {0, 1, 2, 3, 9, 13}

A ∩ B - contains all the common elements in set A and B.

A ∩ B = {0, 1, 3, 4} ∩ {5, 6, 1, 3, 9} = {1, 3}

(A ∩ B) ∪ C - contains all the elements in set A ∩ B and C.

(A ∩ B) ∪ C = {1, 3} ∪ {0, 1, 2, 3, 9, 13}

(A ∩ B) ∪ C = {0, 1, 2, 3, 9, 13}

Hence, option 1 is the correct option.

Let A = {1, 2, 3, 4, 5}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}. Then, (A ∪ B) ∩ C is

1. {3, 4}

2. {3, 4, 5, 6}

3. {1, 2, 3, 4, 5, 6, 8}

4. {4, 6}

Answer

A = {1, 2, 3, 4, 5}

B = {2, 4, 6, 8}

C = {3, 4, 5, 6}

A ∪ B - contains all the elements in set A and B.

A ∪ B = {1, 2, 3, 4, 5} ∪ {2, 4, 6, 8} = {1, 2, 3, 4, 5, 6, 8}

(A ∪ B) ∩ C - contains all the common elements in set A ∪ B and C.

(A ∪ B) ∩ C = {1, 2, 3, 4, 5, 6, 8} ∩ {3, 4, 5, 6}

(A ∪ B) ∩ C = {3, 4, 5, 6}

Hence, option 2 is the correct option.

If A has 3 elements and B has 6 elements, then the minimum and maximum number of elements in A ∪ B are respectively

1. 3 and 6

2. 3 and 9

3. 6 and 9

4. Cannot be determined

Answer

Here n(A) = 3 and n(B) = 6.

A ∪ B contains all the elements in set A and B, so A ∪ B cannot have fewer elements than B.

The number of elements in A ∪ B is the least when every element of A is also an element of B, i.e., when A ⊂ B.

In that case, A ∪ B = B and so n(A ∪ B) = 6.

The number of elements in A ∪ B is the greatest when A and B have no element in common.

In that case, n(A ∪ B) = n(A) + n(B) = 3 + 6 = 9.

So, the minimum and maximum number of elements in A ∪ B are 6 and 9 respectively.

Hence, option 3 is the correct option.

If A has 3 elements and B has 6 elements, then the minimum and maximum number of elements in A ∩ B are respectively

1. 0 and 3

2. 0 and 6

3. 3 and 6

4. 3 and 9

Answer

Here n(A) = 3 and n(B) = 6.

A ∩ B contains all the common elements in set A and B.

The number of elements in A ∩ B is the least when A and B have no element in common.

In that case, A ∩ B = φ and so n(A ∩ B) = 0.

The number of elements in A ∩ B is the greatest when every element of A is also an element of B, i.e., when A ⊂ B.

In that case, A ∩ B = A and so n(A ∩ B) = 3.

So, the minimum and maximum number of elements in A ∩ B are 0 and 3 respectively.

Hence, option 1 is the correct option.

Consider the following statements:

1. (A ∪ B)′ = A′ ∪ B′

2. (φ′)′ = ξ

3. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

4. ξ′ = φ

Of these statements

1. 2 and 3 are correct

2. 1 and 3 are correct

3. 2 and 4 are correct

4. 3 and 4 are correct

Answer

By De-Morgan's law, (A ∪ B)′ = A′ ∩ B′ and not A′ ∪ B′.

∴ Statement 1 is not correct.

The complement of the empty set is the universal set, so φ′ = ξ.

∴ (φ′)′ = ξ′ = φ ≠ ξ

∴ Statement 2 is not correct.

By the distributive law, A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

∴ Statement 3 is correct.

The complement of the universal set is the empty set, so ξ′ = φ.

∴ Statement 4 is correct.

So, statements 3 and 4 are correct.

Hence, option 4 is the correct option.

If A ⊆ B, then A ∩ B is equal to

1. A

2. B

3. φ

4. None of these

Answer

A ∩ B - contains all the common elements in set A and B.

As A ⊆ B, every element of A is also an element of B.

So, the elements common to A and B are exactly the elements of A.

∴ A ∩ B = A

Hence, option 1 is the correct option.

If A ⊆ B, then A ∪ B is equal to

1. A

2. B

3. φ

4. None of these

Answer

A ∪ B - contains all the elements in set A and B.

As A ⊆ B, every element of A is also an element of B.

So, A ∪ B contains no element other than the elements of B.

∴ A ∪ B = B

Hence, option 2 is the correct option.

Directions (Question 22 to 26)
Answer these questions on the basis of the following information:

Consider the following sets:

A = {x : x ∈ N, x is a factor of 3}

B = {x : x ∈ N, x is a factor of 5}

C = {x : x ∈ N, x is a factor of 9}

A ∩ B = ?

1. {x : x ∈ N, x is a factor of 3 or 5}

2. {x : x ∈ N, x is a factor of both 3 and 5}

3. {x : x ∈ N, x is a factor of 15}

4. Both 2 and 3

Answer

A = {x : x ∈ N, x is a factor of 3}

A = {1, 3}

B = {x : x ∈ N, x is a factor of 5}

B = {1, 5}

A ∩ B - contains all the common elements in set A and B.

A ∩ B = {1, 3} ∩ {1, 5} = {1}

So, A ∩ B contains those natural numbers which are factors of 3 as well as of 5, i.e., factors of both 3 and 5.

As {x : x ∈ N, x is a factor of 15} = {1, 3, 5, 15} ≠ {1}, option 3 is not correct.

Hence, option 2 is the correct option.

A ∪ B = ?

1. {x : x ∈ N, x is a factor of either 3 or 5}

2. {x : x ∈ N, x is a factor of 3 or 5 or both}

3. {x : x ∈ N, x is a factor of 15}

4. {x : x ∈ N, x is a factor of both 3 and 5}

Answer

A = {x : x ∈ N, x is a factor of 3}

A = {1, 3}

B = {x : x ∈ N, x is a factor of 5}

B = {1, 5}

A ∪ B - contains all the elements in set A and B.

A ∪ B = {1, 3} ∪ {1, 5} = {1, 3, 5}

A ∪ B contains those natural numbers, each one of which is either a factor of 3 or a factor of 5 or a factor of both.

As {x : x ∈ N, x is a factor of 15} = {1, 3, 5, 15} ≠ {1, 3, 5}, option 3 is not correct.

Hence, option 2 is the correct option.

A ∪ C = ?

1. {x : x ∈ N, x is a factor of 3}

2. {x : x ∈ N, x is a factor of 9}

3. {x : x ∈ N, x is a factor of 27}

4. None of these

Answer

A = {x : x ∈ N, x is a factor of 3}

A = {1, 3}

C = {x : x ∈ N, x is a factor of 9}

C = {1, 3, 9}

A ∪ C - contains all the elements in set A and C.

A ∪ C = {1, 3} ∪ {1, 3, 9} = {1, 3, 9}

∴ A ∪ C = {x : x ∈ N, x is a factor of 9}

Hence, option 2 is the correct option.

A ∩ C = ?

1. {x : x ∈ N, x is a factor of 3}

2. {x : x ∈ N, x is a factor of 9}

3. {x : x ∈ N, x is a factor of 27}

4. None of these

Answer

A = {x : x ∈ N, x is a factor of 3}

A = {1, 3}

C = {x : x ∈ N, x is a factor of 9}

C = {1, 3, 9}

A ∩ C - contains all the common elements in set A and C.

A ∩ C = {1, 3} ∩ {1, 3, 9} = {1, 3}

∴ A ∩ C = {x : x ∈ N, x is a factor of 3}

Hence, option 1 is the correct option.

How many of the following statements are true:

I. A ⊂ B    II. A ⊆ C    III. B ⊆ A    IV. C ⊆ A

1. Nil

2. One

3. Two

4. All

Answer

A = {x : x ∈ N, x is a factor of 3} = {1, 3}

B = {x : x ∈ N, x is a factor of 5} = {1, 5}

C = {x : x ∈ N, x is a factor of 9} = {1, 3, 9}

I. As 3 ∈ A but 3 ∉ B, so A ⊄ B and statement I is not true.

II. Each of 1 and 3 is an element of C, so A ⊆ C and statement II is true.

III. As 5 ∈ B but 5 ∉ A, so B ⊄ A and statement III is not true.

IV. As 9 ∈ C but 9 ∉ A, so C ⊄ A and statement IV is not true.

So, only one of the given statements is true.

Hence, option 2 is the correct option.

Fill in the blanks:

(i) If A ⊆ B and B ⊆ A, then ________.

(ii) If A = {1, 2, 3} and B = {1, 2, 3, 4}, then A - B = ________.

(iii) If ξ = {a, b, c, d, e, f} and A = {b, d, e}, then A′ = ________.

(iv) x ∈ A ∪ B, ⇒ x ∈ A ________ x ∈ B (and/or)

(v) If A is a subset of B, then B is a ________ of A.

(vi) The set of all possible subsets of a set is called its ________.

Answer

(i) If A ⊆ B and B ⊆ A, then A = B.

As every element of A is in B and every element of B is in A, the two sets are equal.

(ii) If A = {1, 2, 3} and B = {1, 2, 3, 4}, then A - B = φ.

A - B = {1, 2, 3} - {1, 2, 3, 4} and there is no element of A which is not in B.

(iii) If ξ = {a, b, c, d, e, f} and A = {b, d, e}, then A′ = {a, c, f}.

A′ = ξ - A = {a, b, c, d, e, f} - {b, d, e} = {a, c, f}

(iv) x ∈ A ∪ B, ⇒ x ∈ A or x ∈ B (and/or)

A ∪ B is the set of all those elements, each one of which is either in A or in B or in both A and B.

(v) If A is a subset of B, then B is a superset of A.

If A ⊆ B, we write B ⊇ A and say that B is a superset of A.

(vi) The set of all possible subsets of a set is called its power set.

The power set of a set A is denoted by P(A).

Write true (T) or false (F):

(i) Set of all persons on earth is an infinite set ________.

(ii) {x : x + 4 = 4} is a singleton set. ________

(iii) The power set of a set containing 3 elements, has 6 elements. ________

(iv) Two sets A and B are said to be equal if n(A) = n(B). ________

(v) Equal sets are always equivalent ________.

(vi) The minimum number of subsets a non-empty set has is 2. ________

Answer

(i) False

Although the persons on earth are very large in number, we know that the process of counting them will surely come to an end.

So, the set of all persons on earth is a finite set.

(ii) True

⇒ x + 4 = 4

⇒ x = 4 - 4

⇒ x = 0

So, {x : x + 4 = 4} = {0}, which contains exactly one element and hence it is a singleton set.

(iii) False

If a set contains n elements, then its power set contains 2ⁿ elements.

Here n = 3, so the power set has 2³ = 8 elements and not 6.

(iv) False

If n(A) = n(B), then A and B are equivalent sets.

Two sets A and B are equal only if every element of A is in B and every element of B is in A.

For example, if A = {1, 2} and B = {a, b}, then n(A) = n(B) but A ≠ B.

(v) True

If A = B, then A and B have exactly the same elements and so n(A) = n(B).

∴ A ↔ B

(vi) True

A non-empty set A has at least one element.

The empty set is a subset of every set and every set is a subset of itself.

So, φ and A are two different subsets of A, i.e., A has at least 2 subsets.

Assertion (A): If A = {x : x is a letter of the word GEOLOGY} and B = {x : x is a factor of 16}, then A and B are equivalent.

Reason (R): The intersection of two equivalent sets is an empty set.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

2. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).

3. Assertion (A) is true but Reason (R) is false.

4. Assertion (A) is false but Reason (R) is true.

Answer

A = {x : x is a letter of the word GEOLOGY}

The letters of the word GEOLOGY are G, E, O, L, O, G, Y.

Writing each letter once and only once, we get

A = {G, E, O, L, Y}

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

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

16 = 1 × 16 = 2 × 8 = 4 × 4

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

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

As n(A) = n(B), A and B are equivalent sets.

∴ Assertion (A) is true.

Two equivalent sets need only have the same number of elements, so they may well have elements in common.

For example, {1, 2} and {2, 3} are equivalent sets, but their intersection is {2}, which is not an empty set.

∴ Reason (R) is false.

Hence, option 3 is the correct option.

Assertion (A): The complement of an empty set does not exist.

Reason (R): The intersection of a set and its complement is a null set.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

2. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).

3. Assertion (A) is true but Reason (R) is false.

4. Assertion (A) is false but Reason (R) is true.

Answer

The complement of a set A is the set of all those elements of ξ which are not in A.

As every element of ξ is not in φ, the complement of the empty set is the universal set, i.e., φ′ = ξ.

So, the complement of an empty set does exist.

∴ Assertion (A) is false.

A′ contains all those elements of ξ which are not in A, so there can be no element common to A and A′.

∴ A ∩ A′ = φ, which is a null set.

∴ Reason (R) is true.

Hence, option 4 is the correct option.

Assertion (A): The set of natural numbers (N) is a sub set of the set of whole numbers (W).

Reason (R): 0 is the only whole number which is not a natural number.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

2. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).

3. Assertion (A) is true but Reason (R) is false.

4. Assertion (A) is false but Reason (R) is true.

Answer

N = {1, 2, 3, 4, 5, ....} and W = {0, 1, 2, 3, 4, 5, ....}

Every natural number is a whole number, so every element of N is in W.

∴ N ⊆ W

∴ Assertion (A) is true.

The whole numbers are 0, 1, 2, 3, 4, 5, .... and out of these only 0 is not a natural number.

∴ Reason (R) is true.

As W consists of all the natural numbers together with 0 alone, every element of N is in W and hence N ⊆ W.

So, Reason (R) is the correct explanation of Assertion (A).

Hence, option 1 is the correct option.