Describe the following sets in set-builder form:

(i) A = {1, 2, 3, 4, 5, 6}

{x : x ∈ N, x<7}

This is read as x is such that x belongs to natural number and x is less than 7. It satisfies all condition of roster form.

(ii) B =\( \{1, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}, \dfrac{1}{5}, …\}\)

{x : x = \( \dfrac{1}{n}\), n ∈ N}

This is read as x is such that x =\( \dfrac{1}{n}\), where n ∈ N.

(iii) C = {0, 3, 6, 9, 12, ….}

{x : x = 3n, n ∈ Z⁺, the set of positive integers}

This is read as x is such that C is the set of multiples of 3 including 0.

(iv) D = {10, 11, 12, 13, 14, 15}

{x : x ∈ N, 9

This is read as x is such that D is the set of natural numbers which are more than 9 but less than 16.

(v) E = {0}

{x : x = 0}

This is read as x is such that E is an integer equal to 0.

(vi) {1, 4, 9, 16,…, 100}

Where,

1² = 1

2² = 4

3² = 9

4² = 16

10² = 100

So, above set can be expressed in set-builder form as {x²: x ∈ N, 1≤ x ≤10}

(vii) {2, 4, 6, 8,….}

{x: x = 2n, n ∈ N}

This is read as x is such that the given set are multiples of 2.

(viii) {5, 25, 125, 625}

Where,

5¹ = 5

5² = 25

5³ = 125

5⁴ = 625

So, above set can be expressed in set-builder form as {5ⁿ: n ∈ N, 1≤ n ≤ 4}