Sets — Competency Focused Questions

Which of the following is a singleton set?

1. {x | x² = 9, x ∈ Z}

2. {x | x is a prime number and x is even}

3. {x | x = 7 - 7, x ∈ N}

4. {x | x is a vowel in the word "prime"}

Answer

Checking all options:

1. {x | x² = 9, x ∈ Z} ⇒ x = 3 or x = -3, so the set is {-3, 3} — two elements.

2. {x | x is a prime number and x is even} ⇒ the only even prime is 2, so the set is {2} — one element (singleton).

3. {x | x = 7 - 7, x ∈ N} ⇒ x = 0, but 0 ∉ N, so the set is { } — empty.

4. {x | x is a vowel in the word "prime"} ⇒ the vowels are i and e, so the set is {i, e} — two elements.

Hence, option 2 is the correct option.

Which of the following sets is an empty set?

1. {x | x is a positive integer and x < 0}

2. {x | x² = 4, x ∈ Z}

3. {x | x is a letter in the word "apple"}

4. {x | x is a natural number and x is odd}

Answer

Checking all options:

1. {x | x is a positive integer and x < 0} ⇒ no positive integer is less than 0, so the set is { } — empty.

2. {x | x² = 4, x ∈ Z} ⇒ x = 2 or x = -2, so the set is {-2, 2} — not empty.

3. {x | x is a letter in the word "apple"} = {a, p, l, e} — not empty.

4. {x | x is a natural number and x is odd} = {1, 3, 5, ......} — not empty.

Hence, option 1 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 prime numbers are 3, 5, 7, 11 and 13.

So, A = {3, 5, 7, 11, 13}, which has 5 elements i.e. n(A) = 5.

Hence, option 3 is the correct option.

Two sets are defined as

A = {x | x is an even prime numbers less than 100}

B = {x | x is an odd composite number less than 100}

Which of the following numbers is common in both?

1. 29

2. 53

3. 97

4. none of these

Answer

A = {x | x is an even prime numbers less than 100}. The only even prime number is 2, so A = {2}.

B = {x | x is an odd composite number less than 100} = {9, 15, 21, 25, 27, ......}, which contains only odd numbers.

Since A contains only the even number 2 and B contains only odd numbers, there is no element common to both. (Also, 29, 53 and 97 are all prime numbers, so none of them belongs to A or B.)

Hence, option 4 is the correct option.

Two sets are defined as A = {1, 2, 3} and B = {3, 2, 1}. What can be concluded?

1. A ≠ B, because their orders are different

2. A = B, because they contain the same elements

3. A is a subset of B, but B is not a subset of A.

4. A and B are finite but not equal sets.

Answer

The order in which the elements of a set are listed is immaterial. Both A and B contain exactly the same elements 1, 2 and 3.

So, A = B, because they contain the same elements.

Hence, option 2 is the correct option.

Which of the following sets is finite?

1. {x | x > 0, x ∈ N}

2. {x | x² < 100, x ∈ N}

3. {x | x is a prime number}.

4. {x | x ≥ 0, x ∈ Z}

Answer

Checking all options:

1. {x | x > 0, x ∈ N} = {1, 2, 3, ......} — infinite.

2. {x | x² < 100, x ∈ N}. Here x² < 100 ⇒ x < 10, so x = 1, 2, 3, ......, 9. The set is {1, 2, 3, 4, 5, 6, 7, 8, 9} — finite.

3. {x | x is a prime number} = {2, 3, 5, 7, 11, ......} — infinite.

4. {x | x ≥ 0, x ∈ Z} = {0, 1, 2, 3, ......} — infinite.

Hence, option 2 is the correct option.