Q.Match each of the set on the left in the roster form with the same set on the right described in set-builder form:

Concept: Set Membership

We match each roster-form set with its set-builder description by checking which elements satisfy the given condition.

1. {1, 2, 3, 6}: The natural divisors of 6 are exactly 1, 2, 3, and 6. This matches (c) ${x : x \text{ is a natural number and divisor of } 6}$.
2. {2, 3}: The prime divisors of 6 are 2 and 3 (since $6 = 2 \times 3$). This matches (a) ${x : x \text{ is a prime number and a divisor of } 6}$. …

The heart of this problem is recognizing that the same set can be written in two ways. Roster form lists every element explicitly, while set-builder form describes the rule or property that determines membership. To match them, we need to verify that both forms capture exactly the same collection of objects—no more, no fewer.

Let's examine each roster-form set and find its set-builder partner.

Matching Process

1. Set (i): $\{1, 2, 3, 6\}$

What do these numbers have in common? They're all natural numbers that divide 6 evenly. Check: $6 \div 1 = 6$, $6 \div 2 = 3$, $6 \div 3 = 2$, $6 \div 6 = 1$. These are precisely the divisors of 6.

Now scan the right column. Option (c) says "$x$ is a natural number and divisor of 6." The divisors of 6 are exactly $1, 2, 3, 6$. Perfect match.

Could it be (a)? No—(a) requires $x$ to be prime and a divisor of 6. The prime divisors of 6 are only 2 and 3, not the full set.

Match: (i) ↔ (c)

2. Set (ii): $\{2, 3\}$

Both elements are prime numbers. Are they related to 6? Yes—they're the only primes that divide 6. ($6 = 2 \times 3$, and 1 is not prime.)

Option (a) states "$x$ is a prime number and a divisor of 6." The prime factorization of 6 is $2 \times 3$, so the prime divisors are exactly 2 and 3.

Match: (ii) ↔ (a)

3. Set (iii): $\{M, A, T, H, E, I, C, S\}$

These are letters, not numbers. The word MATHEMATICS contains the letters M-A-T-H-E-M-A-T-I-C-S. When we write a set, we list each distinct element once (sets don't have duplicates). The distinct letters are: M, A, T, H, E, I, C, S—exactly eight letters.

Option (d) says "$x$ is a letter of the word MATHEMATICS." This captures all distinct letters in that word. …

Method: Set Membership Verification by Element Matching

This method works by checking whether every element of the roster-form set satisfies the condition in the set-builder form, and vice versa.

Steps:

1. List the elements of the roster-form set clearly.
2. Read the condition in the set-builder form carefully.
3. Check each element of the roster set against the condition:
   • If all elements satisfy the condition → possible match.
   • If any element fails → not a match.
4. Also check that no extra elements satisfying the condition are missing from the roster set (for complete matching).

Applying the steps:

(i) {1, 2, 3, 6}

• Check (a): prime divisors of 6 → {2, 3} only. 1 and 6 are not prime → not a match.
• Check (c): natural numbers that divide 6 → {1, 2, 3, 6} → matches exactly.

Match: (i) → (c)

(ii) {2, 3}

• Check (a): prime divisors of 6 → {2, 3} → matches exactly.

Match: (ii) → (a)

(iii) {M, A, T, H, E, I, C, S}

• Check (d): letters of the word MATHEMATICS → {M, A, T, H, E, I, C, S} (note: M, A, T repeat but set removes duplicates) → matches exactly.

Match: (iii) → (d) …

🚩 Mistake 1: Confusing "divisor of 6" with "prime divisor of 6"

The error:

Students match {1, 2, 3, 6} with (a) because they see 2 and 3 are prime divisors, but forget that (a) says only prime divisors.

How to avoid:

• Read the entire condition in set-builder form.
• (a) says: $x$ is a prime number and a divisor of 6 → only {2, 3}.
• (c) says: $x$ is a natural number and divisor of 6 → {1, 2, 3, 6}.

✓ Correct match:

• (i) {1, 2, 3, 6} → (c)
• (ii) {2, 3} → (a)

🚩 Mistake 2: Forgetting that 1 is not prime

The error:

Some students think 1 is prime and match {1, 2, 3, 6} with (a).

How to avoid:

• Memorise: 1 is neither prime nor composite.
• Prime numbers start from 2.

🚩 Mistake 3: Miscounting letters in "MATHEMATICS"

The error:

Students list {M, A, T, H, E, M, A, T, I, C, S} (11 letters) instead of the set of distinct letters.

How to avoid:

• A set has no repetitions.
• Write the word, then remove duplicates: M, A, T, H, E, I, C, S → 8 distinct letters.

✓ Correct match:

• (iii) {M, A, T, H, E, I, C, S} → (d)

🚩 Mistake 4: Misreading "odd natural number less than 10"

The error:

Students include 0 or forget that natural numbers start from 1. Some also include 2 (even) or 10 (not less than 10).

How to avoid:

• Natural numbers: $\{1, 2, 3, \dots\}$ (usually in NCERT).
• "Odd" → only 1, 3, 5, 7, 9.
• "Less than 10" → excludes 10.

✓ Correct match: …