Sets — Multiple Choice Questions

A set is a

1. collection of similar objects

2. collection of different objects

3. collection of same type of objects

4. well defined collection of objects

Answer

A set is a well-defined collection of objects i.e. a collection for which it is possible to tell beyond doubt whether a given object belongs to it or not.

Hence, option 4 is the correct option.

In Roster form, we have φ = ?

1. { }

2. ( )

3. [ ]

4. none of these

Answer

The empty set has no element at all. In roster form, it is written as { }.

Hence, option 1 is the correct option.

n(φ) = ?

1. 1

2. 2

3. 0

4. none of these

Answer

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

Hence, option 3 is the correct option.

A set having only one element is called

1. an infinite set

2. an empty set

3. a singleton set

4. none of these

Answer

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

Hence, option 3 is the correct option.

If A = {x : x is a letter in the word ENGAGEMENT}, then n(A) = ?

1. 10

2. 8

3. 7

4. 6

Answer

The letters in the word ENGAGEMENT are E, N, G, A, G, E, M, E, N, T. Writing each letter only once, we get E, N, G, A, M, T.

So, A = {E, N, G, A, M, T}, which has 6 elements i.e. n(A) = 6.

Hence, option 4 is the correct option.

Which of the following is an empty set?

1. {x : x ∈ W, x + 2 = 2}

2. {x : x is a prime number, x < 3}

3. {x : x ∈ I, x + 5 = 3}

4. {x : x ∈ N, x + 8 = 8}

Answer

Checking all options:

1. {x : x ∈ W, x + 2 = 2} ⇒ x = 0. Since 0 ∈ W, the set is {0} — not empty.

2. {x : x is a prime number, x < 3} ⇒ the only prime number less than 3 is 2, so the set is {2} — not empty.

3. {x : x ∈ I, x + 5 = 3} ⇒ x = -2. Since -2 ∈ I, the set is {-2} — not empty.

4. {x : x ∈ N, x + 8 = 8} ⇒ x = 0. Since 0 ∉ N, no natural number satisfies this — an empty set.

Hence, option 4 is the correct option.

Which of the following is a finite set?

1. {x : x ∈ I, x < 3}

2. {x : x ∈ N, x > 50}

3. {x : x ∈ I, x > -10}

4. {x : x ∈ N, 1 < x < 50}

Answer

Checking all options:

1. {x : x ∈ I, x < 3} = {......, -1, 0, 1, 2} — infinite.

2. {x : x ∈ N, x > 50} = {51, 52, 53, ......} — infinite.

3. {x : x ∈ I, x > -10} = {-9, -8, ......, 0, 1, 2, ......} — infinite.

4. {x : x ∈ N, 1 < x < 50} = {2, 3, 4, ......, 49} — finite, as the counting comes to an end.

Hence, option 4 is the correct option.

Which of the following is an infinite set?

1. {x : x ∈ N, x < 500}

2. {x : x ∈ N, x > 500}

3. {x : x ∈ N, 50 < x < 500}

4. {x : x ∈ I, -5000 < x < 5000}

Answer

Checking all options:

1. {x : x ∈ N, x < 500} = {1, 2, 3, ......, 499} — finite.

2. {x : x ∈ N, x > 500} = {501, 502, 503, ......} — infinite, as the counting never comes to an end.

3. {x : x ∈ N, 50 < x < 500} = {51, 52, ......, 499} — finite.

4. {x : x ∈ I, -5000 < x < 5000} = {-4999, ......, 0, ......, 4999} — finite.

Hence, option 2 is the correct option.

If A = {1, 2, 3} and B = {4, 5, 6}, then A and B are

1. equal

2. equivalent

3. empty

4. none of these

Answer

A = {1, 2, 3} and B = {4, 5, 6}, so n(A) = 3 and n(B) = 3.

The two sets have the same cardinal number but different elements. So they are equivalent (not equal).

Hence, option 2 is the correct option.

If A = {x : x ∈ N and x² < 30}, then n(A) = ?

1. 7

2. 6

3. 5

4. 4

Answer

We take natural numbers x for which x² < 30.

For x = 1, x² = 1 < 30 ✓

For x = 2, x² = 4 < 30 ✓

For x = 3, x² = 9 < 30 ✓

For x = 4, x² = 16 < 30 ✓

For x = 5, x² = 25 < 30 ✓

For x = 6, x² = 36 > 30 ✗

So, A = {1, 2, 3, 4, 5}, which has 5 elements i.e. n(A) = 5.

Hence, option 3 is the correct option.