Set Concepts — Multiple Choice Questions

Set A = {0, 3, 6, 9, 12, .....} can be written as :

1. A = {x : x = 3n and n ∈ N}

2. A = {x : x = 2n − 1 and n ∈ W}

3. A = {x : x = 2n + 1 and n ∈ W}

4. A = {x : x = 3n and n ∈ W}

Answer

The elements 0, 3, 6, 9, 12, ... are multiples of 3 starting from 0. So x = 3n where n must include 0, i.e. n ∈ W.

When n ∈ W, x = 3n gives 0, 3, 6, 9, 12, ...

A = {x : x = 3n and n ∈ W}

Hence, Option 4 is the correct option.

{x : x = n² − 1, n ∈ N and n < 5} in roster form is :

1. {−1, 0, 3, 8, 15, 24}

2. {3, 8, 15, 24}

3. {0, 3, 8, 15}

4. {0, 3, 8, 15, 24}

Answer

Here n ∈ N and n < 5, so n = 1, 2, 3, 4.

When n = 1, x = 1² − 1 = 0

When n = 2, x = 2² − 1 = 3

When n = 3, x = 3² − 1 = 8

When n = 4, x = 4² − 1 = 15

So the set is {0, 3, 8, 15}.

Hence, Option 3 is the correct option.

If cardinal number of set A is 8 i.e. n(A) = 8 and cardinal number of set B is also 8 i.e. n(B) = 8; then :

1. set A = set B

2. A and B are equivalent sets

3. none of these

Answer

Since n(A) = n(B) = 8, the two sets have the same number of elements, so they are equivalent. They need not have the same elements, so they need not be equal.

A and B are equivalent sets

Hence, Option 2 is the correct option.

If set A = {x : x = n³, n ∈ W and n < 3}, the number of subsets of set A are :

1. 2

2. 2³

3. 2²

4. 2⁴

Answer

Here n ∈ W and n < 3, so n = 0, 1, 2.

When n = 0, x = 0³ = 0

When n = 1, x = 1³ = 1

When n = 2, x = 2³ = 8

So A = {0, 1, 8}, which has 3 elements.

Number of subsets = 2³.

Hence, Option 2 is the correct option.

A = {letters of word JANTAR} and B = {letters of word AJANTA}, then :

1. A = B

2. A ⊂ B

3. B ⊂ A

4. none of these

Answer

A = {letters of JANTAR} = {J, A, N, T, R}

B = {letters of AJANTA} = {A, J, N, T}

Every element of B is in A, and B ≠ A (since R ∈ A but R ∉ B). So B is a proper subset of A.

B ⊂ A

Hence, Option 3 is the correct option.

{x : x ∈ I (integer) and x² < 16} in roster form is :

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

2. {0, 1, 2, 3}

3. {−3, −2, −1, 0, 1, 2, 3}

4. {−4, −3, −2, −1, 0, 1, 2, 3, 4}

Answer

We need integers x with x² < 16, i.e. −4 < x < 4.

The integers satisfying this are −3, −2, −1, 0, 1, 2 and 3 (note that (±4)² = 16, which is not less than 16).

So, the roster form is {−3, −2, −1, 0, 1, 2, 3}.

Hence, Option 3 is the correct option.

If set A = {7, 8, 9, 10} and set B = {0, 1, 2, 3} then A − B is equal to :

1. {7, 7, 7, 7} = {7}

2. {7, 8, 9, 10}

3. {0, 1, 2, 3}

4. none of these

Answer

A − B = {elements of A which are not in B}.

Since A and B have no common element, all elements of A remain.

So A − B = {7, 8, 9, 10}.

Hence, Option 2 is the correct option.

If universal set S = {x : x ∈ Z (integers) and −2 < x ≤ 2} and set A = {−1, 0, 1} then complement of set A i.e. A′ is equal to :

1. {−2, 2}

2. {2}

3. {−1, 0, 1}

4. φ

Answer

S = {x : x ∈ Z and −2 < x ≤ 2} = {−1, 0, 1, 2}

A = {−1, 0, 1}

A′ = S − A = {elements of S which are not in A} = {2}

Hence, Option 2 is the correct option.

A set has 5 elements, then number of its proper subsets is :

1. 2⁵

2. 2⁵ − 1

3. 2^{5 − 1}

4. 2 × n

Answer

If a set has n elements, the number of its proper subsets is 2ⁿ − 1.

Here n = 5, so the number of proper subsets = 2⁵ − 1.

Hence, Option 2 is the correct option.

If set A = {4, 5, 6, 7}, then a proper subset of set A is :

1. {6, 7, 8}

2. {3, 4}

3. {8}

4. { }

Answer

A proper subset of A must contain only elements of A and must not be equal to A.

{6, 7, 8} contains 8 ∉ A; {3, 4} contains 3 ∉ A; {8} contains 8 ∉ A. So these are not subsets of A.

The empty set { } is a proper subset of every non-empty set.

Hence, Option 4 is the correct option.

If set A = {0, 1, 2, 3} and set B = {6, 7, 8, 9} then A ∪ B is equal to :

1. {6, 8, 10, 12}

2. {0, 1, 2, 3, 6, 7, 8, 9}

3. { }

4. {0, 1, 2, 3}

Answer

A ∪ B = {all elements of A and of B} = {0, 1, 2, 3, 6, 7, 8, 9}.

Hence, Option 2 is the correct option.

If set A = {0, 1, 2, 3} and set B = {6, 7, 8, 9}, then B ∩ A is equal to :

1. {0, 1, 2, 3, 6, 7, 8, 9}

2. {−6, −6, −6, −6} = {−6}

3. { }

4. none of these

Answer

B ∩ A = {elements common to A and B}. Since A and B have no common element, B ∩ A = { }.

Hence, Option 3 is the correct option.

For any two sets A and B :

1. n(A) + n(B) = n(A ∪ B)

2. n(A) − n(B) = n(A − B)

3. n(A ∪ B) = n(A) + n(B) + n(A ∩ B)

4. n(A ∪ B) + n(A ∩ B) = n(A) + n(B)

Answer

For any two sets A and B, the correct cardinal property is

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

which can be rearranged as n(A ∪ B) + n(A ∩ B) = n(A) + n(B).

Hence, Option 4 is the correct option.

If number of proper subsets of a set is 2⁴ − 1, the number of elements in the set is :

1. 3

2. 5

3. 2

4. 4

Answer

The number of proper subsets of a set with n elements is 2ⁿ − 1.

Given 2ⁿ − 1 = 2⁴ − 1, so n = 4.

Hence, Option 4 is the correct option.

If for sets A, B and C, A ⊆ B and B ⊆ C, then :

1. A = C

2. C ⊆ A

3. A ⊄ C

4. A ⊆ C

Answer

If every element of A is in B (A ⊆ B) and every element of B is in C (B ⊆ C), then every element of A is in C.

So A ⊆ C.

Hence, Option 4 is the correct option.