Exercise 2.1 — Roster and Set Builder

Problems based on roster form and set builder form.

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Exercise 2.1 — Sets (Class 10 Mathematics)

Exercise 2.1 from Chapter 2, Sets, is part of the CBSE, Telangana SSC, and Andhra Pradesh SSC Class 10 Mathematics syllabus. This exercise tests your understanding of the foundational ideas from the chapter: identifying whether a collection is a well-defined set, using the membership symbols ∈ and ∉, and converting freely between roster form and set-builder form.

These eight questions cover everything you need for board-exam-style set theory problems — from simple true/false justifications to matching exercises that test your ability to read sets both ways.

Well-Defined Collections ∈ and ∉ Notation Roster Form Set-Builder Form True/False Justification Matching Sets

💡 Before you begin: Remember — a collection is a set only if there is a clear, unambiguous rule to decide whether any given object belongs to it. Vague words like "best," "talented," or "tall" make a collection not well defined.

Question 1 — Which of the Following Are Sets?

For each collection below, we must justify why it is or is not a well-defined set.

Q1 (i)

The collection of all the months of a year beginning with the letter "J".

✅ YES — IT IS A SET

There are exactly three months beginning with "J": January, June, and July. Anyone can check this list and confirm membership without doubt — so the collection is well defined.

Q1 (ii)

The collection of ten most talented writers of India.

❌ NO — NOT A SET

"Talented" is subjective. There is no fixed rule or measurable criterion to decide whether a particular writer belongs to this "top ten" — different people would give different lists. Hence it is not well defined, and not a set.

Q1 (iii)

A team of eleven best cricket batsmen of the world.

❌ NO — NOT A SET

Just like "talented," the word "best" has no precise, universally agreed-upon rule. Different experts would name different players. Since membership cannot be determined with certainty, this collection is not a set.

Q1 (iv)

The collection of all boys in your class.

✅ YES — IT IS A SET

Whether a particular student is a boy in your class is a fact that can be checked with certainty — there's no ambiguity. So this collection is well defined and qualifies as a set.

Q1 (v)

The collection of all even integers.

✅ YES — IT IS A SET

Every integer is either even or odd — this can always be checked precisely (divisibility by 2). The even integers are 2, 4, 6, 8, … and this collection is clearly well defined.

📌 Pattern to notice: Q1(ii) and Q1(iii) fail for the exact same reason — they both depend on subjective judgement ("talented," "best") rather than a fixed, checkable rule. Whenever a problem uses opinion-based words, the answer is almost always "not a set."

Question 2 — Fill in ∈ or ∉

Given: A = {0, 2, 4, 6}, B = {3, 5, 7}, C = {p, q, r}. Fill the appropriate symbol in each blank.

Statement	Symbol	Reasoning
(i) 0 ___ A	∈	0 is an element of A = {0, 2, 4, 6}
(ii) 3 ___ C	∉	3 is a number; C = {p, q, r} contains only letters
(iii) 4 ___ B	∉	4 is not in B = {3, 5, 7}
(iv) 8 ___ A	∉	8 is not in A = {0, 2, 4, 6}
(v) p ___ C	∈	p is an element of C = {p, q, r}
(vi) 7 ___ B	∈	7 is an element of B = {3, 5, 7}

💡 Quick check method: To decide ∈ or ∉, simply scan the listed elements of the set and see if the given item appears exactly as written. If yes → ∈. If no → ∉.

Question 3 — Express Statements Using Symbols

Statement	Symbolic Form
(i) The element 'x' does not belong to 'A'	x ∉ A
(ii) 'd' is an element of the set 'B'	d ∈ B
(iii) '1' belongs to the set of natural numbers	1 ∈ N
(iv) '8' does not belong to the set of prime numbers P	8 ∉ P

📌 This question simply tests translation skill — converting plain English sentences about set membership into mathematical symbols. The key words to watch for are "belongs to" (→ ∈) and "does not belong to" (→ ∉).

Question 4 — True or False, with Justification

Q4 (i)

5 ∉ set of prime numbers.

❌ False

5 is a prime number (its only factors are 1 and 5), so 5 actually belongs to the set of prime numbers. The statement claiming "5 does not belong" is therefore false.

Q4 (ii)

S = {5, 6, 7} implies 8 ∈ S.

❌ False

The set S only contains the elements 5, 6, and 7. Since 8 does not appear in S, the claim "8 ∈ S" is false.

Q4 (iii)

−5 ∉ W, where W is the set of whole numbers.

✅ True

Whole numbers are W = {0, 1, 2, 3, …} — they never include negative numbers. So −5 is indeed not in W, making this statement true.

Q4 (iv)

8/11 ∈ Z, where Z is the set of integers.

❌ False

Integers are whole numbers (positive, negative, or zero) with no fractional part: Z = {…, −2, −1, 0, 1, 2, …}. Since 8/11 is a fraction, not a whole number, it does not belong to Z. The statement is false.

⚠️ Number-set confusion is the most common mistake here. Always remember the hierarchy: Natural numbers (N) = {1,2,3,…} ⊂ Whole numbers (W) = {0,1,2,3,…} ⊂ Integers (Z) = {…,−2,−1,0,1,2,…} ⊂ Rational numbers (Q) = fractions and integers. A fraction like 8/11 belongs to Q, not to N, W, or Z.

Question 5 — Convert to Roster Form

Q5 (i)

B = {x : x is a natural number smaller than 6}

Natural numbers less than 6: 1, 2, 3, 4, 5 B = {1, 2, 3, 4, 5}

Q5 (ii)

C = {x : x is a two-digit natural number such that the sum of its digits is 8}

Find all two-digit numbers whose digits add to 8: 17 (1+7=8), 26 (2+6=8), 35 (3+5=8), 44 (4+4=8) 53 (5+3=8), 62 (6+2=8), 71 (7+1=8), 80 (8+0=8) C = {17, 26, 35, 44, 53, 62, 71, 80}

Q5 (iii)

D = {x : x is a prime number which is a divisor of 60}

60 = 2 × 2 × 3 × 5 → prime factors: 2, 3, 5 D = {2, 3, 5}

Q5 (iv)

E = {x : x is an alphabet in BETTER}

Letters in BETTER: B, E, T, T, E, R → distinct letters only E = {B, E, T, R}

Question 6 — Convert to Set-Builder Form

Q6 (i)

{3, 6, 9, 12}

Pattern: each element is a multiple of 3, and all are less than 13.

{x : x is a multiple of 3 and x < 13}

Q6 (ii)

{2, 4, 8, 16, 32}

Pattern: each element is a power of 2 — 2¹, 2², 2³, 2⁴, 2⁵.

{x : x = 2ⁿ, n ∈ N, n ≤ 5}

Q6 (iii)

{5, 25, 125, 625}

Pattern: each element is a power of 5 — 5¹, 5², 5³, 5⁴.

{x : x = 5ⁿ, n ∈ N, n ≤ 4}

Q6 (iv)

{1, 4, 9, 16, 25, …, 100}

Pattern: each element is a perfect square — 1², 2², 3², …, 10².

{x : x = n², n ∈ N, n ≤ 10}

💡 Strategy for set-builder conversion: Look for a mathematical pattern connecting the elements — multiples, powers, squares, primes, or a range. Once you spot the pattern, express it as a rule using a variable like x or n.

Question 7 — Convert to Roster Form

Q7 (i)

A = {x : x is a natural number greater than 50 but smaller than 100}

A = {51, 52, 53, …, 98, 99}

Q7 (ii)

B = {x : x is an integer, x² = 4}

x² = 4 → x = +2 or x = −2 B = {−2, 2}

Q7 (iii)

D = {x : x is a letter in the word "LOYAL"}

Letters in LOYAL: L, O, Y, A, L → distinct letters only D = {L, O, Y, A}

⚠️ Don't forget the negative root! In Q7(ii), many students write B = {2} and forget that (−2)² = 4 as well. Whenever a set-builder condition involves squaring, check for both positive and negative solutions.

Question 8 — Match Roster Form with Set-Builder Form

Roster Form	Matches	Set-Builder Form
(i) {1, 2, 3, 6}	→ (c)	x is a natural number and divisor of 6
(ii) {2, 3}	→ (a)	x is a prime number and a divisor of 6
(iii) {m, a, t, h, e, i, c, s}	→ (d)	x is a letter of the word MATHEMATICS
(iv) {1, 3, 5, 7, 9}	→ (b)	x is an odd natural number smaller than 10

How the Matching Works

(i) → (c): The natural number divisors of 6 are 1, 2, 3, 6 — matches exactly. Note this is broader than (ii), since it includes non-prime divisors too.
(ii) → (a): Among the divisors of 6, only 2 and 3 are prime numbers (1 is not prime, 6 is not prime).
(iii) → (d): The word MATHEMATICS has the distinct letters m, a, t, h, e, i, c, s — exactly 8 letters, matching the roster set.
(iv) → (b): The odd natural numbers below 10 are 1, 3, 5, 7, 9.

✅ i→c, ii→a, iii→d, iv→b

Common Mistakes to Avoid

Confusing "best/talented/tall" with well-defined rules — Any collection based on opinion or subjective ranking is NOT a set, no matter how reasonable the criteria sound.
Forgetting negative solutions — When a set-builder condition involves x² = k, remember both +√k and −√k are solutions (if x is allowed to be any integer).
Mixing up number systems — Always check: is the given number a natural number, whole number, integer, or rational number? A fraction is never in N, W, or Z.
Repeating letters in word-based sets — When listing letters of a word as a set (e.g., LOYAL, BETTER, MATHEMATICS), always remove duplicate letters since sets only contain distinct elements.
Reversing ∈ and ∉ — Double-check direction: ∈ means "is a member of," ∉ means "is NOT a member of." A quick scan of the set's listed elements always confirms which symbol is correct.

Board Exam Tips & What This Lesson Prepares You For

Exercise 2.1 is foundational and highly scoring in Telangana SSC, AP SSC, and CBSE Class 10 Board Exams. Expect direct questions on identifying well-defined sets, filling in ∈/∉, and converting between roster and set-builder forms — these are common 1-mark and 2-mark questions.

For "is this a set?" questions, always give a one-line justification — examiners specifically look for the reasoning, not just yes/no.
Practice spotting number patterns quickly (multiples, powers, squares, primes) — this is the core skill tested in set-builder conversions.
Remember the number system hierarchy: N ⊂ W ⊂ Z ⊂ Q — this resolves most "true or false" questions about set membership.
When converting words to sets, always check for repeated letters before finalizing your roster form answer.

🎯 Keep going: After completing Exercise 2.1, revisit the foundational ideas in Introduction to Sets if needed, then move ahead to Types of Sets (empty set, finite and infinite sets, equal sets) and eventually Operations on Sets (union, intersection, and difference of sets).