Q.Write the following sets in roster form:

Concept: Set Membership — we list only those elements that satisfy the given condition.

(i) Integers from $-3$ up to but not including $7$:

$A = \{-3, -2, -1, 0, 1, 2, 3, 4, 5, 6\}$

(ii) Natural numbers less than $6$ (natural numbers start from $1$):

$B = \{1, 2, 3, 4, 5\}$

(iii) Two-digit natural numbers where digits sum to $8$:

Possible tens digits $1$ to $8$; units digit = $8 - \text{tens digit}$.

$C = \{17, 26, 35, 44, 53, 62, 71, 80\}$

(iv) Prime divisors of $60$: $60 = 2^2 \times 3 \times 5$, so primes are $2, 3, 5$.

$D = \{2, 3, 5\}$

(v) Distinct letters in TRIGONOMETRY: T, R, I, G, O, N, M, E, Y.

$E = \{T, R, I, G, O, N, M, E, Y\}$ …

1. Set A: integers from –3 up to (but not including) 7

The condition says $x$ is an integer and $-3 \le x < 7$.

That means we start at $-3$ and include every integer until we reach $6$ (since $7$ is not included).

List them: $-3, -2, -1, 0, 1, 2, 3, 4, 5, 6$.

So

2. Set B: natural numbers less than 6

Natural numbers are the counting numbers: $1, 2, 3, 4, 5, \dots$

“Less than 6” means $x < 6$, so we take $1, 2, 3, 4, 5$.

Thus

3. Set C: two-digit natural numbers whose digits sum to 8

A two-digit number has a tens digit $t$ (from 1 to 9) and a units digit $u$ (from 0 to 9).

We need $t + u = 8$.

Let’s go through possible tens digits:

• $t = 1 \Rightarrow u = 7$ → number 17
• $t = 2 \Rightarrow u = 6$ → 26
• $t = 3 \Rightarrow u = 5$ → 35
• $t = 4 \Rightarrow u = 4$ → 44
• $t = 5 \Rightarrow u = 3$ → 53
• $t = 6 \Rightarrow u = 2$ → 62
• $t = 7 \Rightarrow u = 1$ → 71
• $t = 8 \Rightarrow u = 0$ → 80
• $t = 9 \Rightarrow u = -1$ → not possible

So the numbers are: 17, 26, 35, 44, 53, 62, 71, 80.

Therefore

4. Set D: prime divisors of 60

First, find all divisors of 60. Then pick only the prime ones.

Divisors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

From these, the primes are: 2, 3, 5.

Thus

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Here is the Set-Builder to Roster Form method, applied step-by-step to each part.

Method Name: Translation Method (Set-Builder to Roster Form)

Core Idea: Read the condition (the property after the colon) and list every element that satisfies it, separated by commas, inside curly braces { }. Stop when no more elements fit the rule.

(i) A = {x : x is an integer and –3 ≤ x < 7}

Step 1: Identify the type of number: Integer (…, -2, -1, 0, 1, 2, …).

Step 2: Identify the range: from -3 (included, because ≤) up to but not including 7 (because <).

Step 3: List all integers in that range:

-3, -2, -1, 0, 1, 2, 3, 4, 5, 6

Answer:

A = { –3, –2, –1, 0, 1, 2, 3, 4, 5, 6 }

(ii) B = {x : x is a natural number less than 6}

Step 1: Identify the type: Natural number (usually 1, 2, 3, … in Indian exams).

Step 2: Condition: less than 6 → 1, 2, 3, 4, 5.

Answer:

B = { 1, 2, 3, 4, 5 }

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

Step 1: Type: Two-digit natural number (10 to 99).

Step 2: Condition: Sum of digits = 8.

Step 3: Find all pairs (tens digit, units digit) where tens digit ≥ 1 and sum = 8:

• (1,7) → 17
• (2,6) → 26
• (3,5) → 35
• (4,4) → 44
• (5,3) → 53
• (6,2) → 62
• (7,1) → 71
• (8,0) → 80

Answer:

C = { 17, 26, 35, 44, 53, 62, 71, 80 }

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

Step 1: Find all divisors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

Step 2: Pick only the prime numbers from that list:

• 2 (prime)
• 3 (prime)
• 5 (prime) …

(i) $A = \{ x : x \text{ is an integer and } -3 \le x < 7 \}$

✓ Correct Roster Form:

✗ Common Mistakes:

• Forgetting negative integers: Students often start from $0$ or $1$, missing $-3, -2, -1$.
• Including 7: The condition is $x < 7$, so 7 is not included.
• Confusing integer with natural number: Some write only $\{1,2,3,4,5,6\}$.

✓ How to Avoid:

• Read the inequality carefully: ≤ means include, < means exclude.
• Always list the smallest integer first.
• Double-check: "integer" includes negatives, zero, and positives.

(ii) $B = \{ x : x \text{ is a natural number less than 6} \}$

✓ Correct Roster Form:

✗ Common Mistakes:

• Including 0: In Indian NCERT, natural numbers start from 1 (not 0).
• Including 6: "Less than 6" means $x < 6$, so 6 is excluded.

✓ How to Avoid:

• Memorise: In Class 11 NCERT, $\mathbb{N} = \{1,2,3,\dots\}$.
• Write the condition as $x < 6$ and list only numbers satisfying it.

(iii) $C = \{ x : x \text{ is a two-digit natural number such that the sum of its digits is 8} \}$

✓ Correct Roster Form:

✗ Common Mistakes:

• Missing 80: Many forget that $8+0=8$ and 80 is a two-digit number.
• Including 08: 08 is not a two-digit number; it's just 8.
• Missing order: Listing only a few pairs like $(1,7)$ and $(2,6)$.

✓ How to Avoid:

• Systematically list all pairs of digits $(a,b)$ where $a$ is tens digit (1–9), $b$ is units digit (0–9), and $a+b=8$.
• Write them in increasing order to avoid missing any.

(iv) $D = \{ x : x \text{ is a prime number which is divisor of 60} \}$

✓ Correct Roster Form:

✗ Common Mistakes:

• Including 1: 1 is not prime.
• Including composite divisors like 4, 6, 10, 12, 15, 20, 30, 60.
• Including 7, 11, etc.: They are prime but not divisors of 60.

✓ How to Avoid:

• First find all divisors of 60: $1,2,3,4,5,6,10,12,15,20,30,60$.
• Then pick only the prime ones: $2,3,5$.
• Remember: Prime numbers start from 2.

(v) $E = \text{The set of all letters in the word TRIGONOMETRY}$

✓ Correct Roster Form:

✗ Common Mistakes:

• Repeating letters: Writing T twice or O twice.
• Including H: There is no H in TRIGONOMETRY.
• Wrong order or case: Usually written in capital letters, but order doesn't matter.

✓ How to Avoid: …