Identify whether the given pair consists of equal or equivalent but not equal sets or none :

(i) A = set of letters of the word 'FLOWER'.
    B = set of letters of the word 'FOLLOWER'.

(ii) C = {x : x ∈ N, x + 5 = 6} and D = {x : x ∈ W, x < 1}.

(iii) E = set of first five whole numbers.
      F = set of first five natural numbers.

(iv) G = {a, b, c} and H = {x, y, z}.

(v) J = {x : x ∈ N, x ≠ x} and K = {x : x ∈ N, 6 < x < 7}.

(i)

A = Set of letters of the word 'FLOWER'.
B = Set of letters of the word 'FOLLOWER'.

A = {F, L, O, W, E, R} and B = {F, O, L, W, E, R}

In roster form, repeated letters in 'FOLLOWER' are listed only once.

Both sets contain the same elements {F, L, O, W, E, R}. These are equal sets.

∴ A = B

(ii) C = {x : x ∈ N, x + 5 = 6} and D = {x : x ∈ W, x < 1}.

For Set C:

x + 5 = 6

⇒ x = 6 - 5

⇒ x = 1

Since 1 is a natural number, x = {1}.

C = {1}, n(C) = 1

For Set D:

x < 1

The only whole number less than 1 is 0.

D = {0}, n(D) = 1

n(C) = 1 and n(D) = 1. They have the same number of elements but different elements.

They are equivalent but not equal sets

∴ C ↔ D

(iii)

E = set of first five whole numbers.
F = set of first five natural numbers.

E = {0, 1, 2, 3, 4} and F = {1, 2, 3, 4, 5}

E contains 0 to 4 (first five whole numbers). F contains 1 to 5 (first five natural numbers).

n(E) = 5 and n(F) = 5. The cardinal numbers are the same, but the elements are different.

They are equivalent but not equal sets

∴ E ↔ F

(iv) G = {a, b, c} and H = {x, y, z}.

Both sets have three distinct elements.

n(G) = 3 and n(H) = 3. The elements are entirely different.

They are equivalent but not equal sets

∴ G ↔ H

(v) J = {x : x ∈ N, x ≠ x} and K = {x : x ∈ N, 6 < x < 7}.

No natural number is unequal to itself. J = { } (Null set).

There is no natural number between 6 and 7. K = { } (Null set).

Both are empty sets and therefore contain exactly the same (zero) elements. These are equal sets.

∴ J = K