State whether the following pairs of sets are equivalent or not :

(i) A = {x : x ∈ N and 11 ≥ 2x - 1} and
    B = {y : y ∈ W and 3 ≤ y ≤ 9}.

(ii) Set of integers and set of natural numbers.

(iii) Set of whole numbers and set of multiples of 3.

(iv) P = {5, 6, 7, 8} and M = {x : x ∈ W and x ≤ 4}.

(i) Not equivalent

Reason

A = {x : x ∈ N and 11 ≥ 2x - 1}

A = {11 ≥ 2x - 1}

A = {11 + 1 ≥ 2x}

A = {12 ≥ 2x}

A = {x ≤ $\dfrac{12}{2}$}

A = {x ≤ 6}

A = {1, 2, 3, 4, 5, 6}

B = {y : y ∈ W and 3 ≤ y ≤ 9}

B = {3, 4, 5, 6, 7, 8, 9}

There are 6 elements in set A and 7 in set B.

(ii) Not equivalent

Reason

Set of integers contain all negative numbers , 0 and positive numbers.

A = {……………, -2, -1, 0, 1, 2,……………}

Set of natural numbers contain all positive numbers.

B = {1, 2, 3, 4, ……………}

(iii) Not equivalent

Reason

Set of whole numbers contains all natural numbers and 0.

X = {0, 1, 2, 3, 4,……………}

Set of multiples of 3 contains all the numbers which are multiple of 3.

Y = {3, 6, 9, 12, 15, 18,……………}

Hence, they contain different number of questions.

(iv) Not equivalent

Reason

P = {5, 6, 7, 8}

M = {x : x ∈ W and x ≤ 4}

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

As P contains 4 elements whereas M contains 5 elements.