Q.List all the elements of the following sets :

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

(i) Odd natural numbers are $1, 3, 5, 7, \dots$

So $A = \{1, 3, 5, 7, 9, \dots\}$ — an infinite set.

(ii) The condition is $-\frac{1}{2} < x < \frac{9}{2}$ with $x$ an integer.

$-\frac{1}{2} < x$ means $x \ge 0$; $x < \frac{9}{2}$ means $x \le 4$.

So $B = \{0, 1, 2, 3, 4\}$.

(iii) $x^2 \le 4$ for integers means $|x| \le 2$, i.e. $x = -2, -1, 0, 1, 2$.

Thus $C = \{-2, -1, 0, 1, 2\}$.

(iv) The word “LOYAL” has letters L, O, Y, A, L. Distinct letters only:

$D = \{\text{L, O, Y, A}\}$.

(v) Months with 31 days: Jan, Mar, May, Jul, Aug, Oct, Dec. The rest (Feb, Apr, Jun, Sep, Nov) do not have 31 days.

So $E = \{\text{February, April, June, September, November}\}$. …

Set-builder notation is a compact way to define a set by a rule. The notation $\{x : \text{condition on } x\}$ means "the set of all $x$ such that the condition holds." The trick is to read the condition precisely — especially the domain (what kind of numbers or objects $x$ is allowed to be) and the inequality or property that $x$ must satisfy.

Let’s go through each part step by step.

1. Set A: odd natural numbers

The condition: $x$ is an odd natural number. Natural numbers are usually taken as $1,2,3,4,\dots$ (positive integers). Odd means not divisible by 2. So the set is all positive odd integers.

Since the set is infinite, we list it using the pattern:

$A = \{1, 3, 5, 7, \dots\}$

2. Set B: integers between $-\frac{1}{2}$ and $\frac{9}{2}$

The condition: $x$ is an integer, and $-\frac{1}{2} < x < \frac{9}{2}$.

First, convert the fractions: $-\frac{1}{2} = -0.5$ and $\frac{9}{2} = 4.5$. So $x$ must be an integer strictly greater than $-0.5$ and strictly less than $4.5$.

The integers that satisfy this are: $0, 1, 2, 3, 4$. But wait — is $-1$ included? No, because $-1 < -0.5$, so it fails. Is $5$ included? No, because $5 > 4.5$.

So $B = \{0, 1, 2, 3, 4\}$.

3. Set C: integers with $x^2 \leq 4$

Condition: $x$ is an integer, and $x^2 \leq 4$.

Solve $x^2 \leq 4$. This means $|x| \leq 2$, i.e., $-2 \leq x \leq 2$. Since $x$ is an integer, the possible values are $-2, -1, 0, 1, 2$.

So $C = \{-2, -1, 0, 1, 2\}$.

4. Set D: letters in the word "LOYAL"

Condition: $x$ is a letter that appears in the word "LOYAL". The word has letters: L, O, Y, A, L. Note that L appears twice, but in a set, each element is listed only once (sets have no duplicates).

So the distinct letters are: L, O, Y, A.

Thus $D = \{L, O, Y, A\}$.

5. Set E: months not having 31 days

Condition: $x$ is a month of a year that does not have 31 days. …

Here is the clear, concept-first solution method for listing elements of a set from a given rule.

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

Core Concept: A set is defined by a property (the rule after the colon :). To list the elements, you must identify the universal pool of objects (e.g., natural numbers, integers, letters) and then test each object to see if it satisfies the property.

Steps for the Method

1. Identify the Domain: Read the description after x : and before the comma. This tells you the type of object you are allowed to pick (e.g., natural numbers, integers, letters).
2. Identify the Condition: Read the rule after the comma. This is the test each object must pass.
3. Generate Candidates: List the first few objects from the domain (starting from the smallest or most obvious).
4. Apply the Filter: Check each candidate against the condition. Keep only those that pass.
5. Write in Roster Form: Place the kept elements inside curly braces { }, separated by commas.

Applying the Method to Each Set

(i) $A = \{x : x \text{ is an odd natural number}\}$

• Domain: Natural numbers ($\mathbb{N}$). Usually, in NCERT/CBSE, natural numbers start from $1$.
• Condition: $x$ must be odd (not divisible by 2).
• Candidates: $1, 2, 3, 4, 5, 6, 7, \dots$
• Filter: Keep $1, 3, 5, 7, \dots$ (all odd numbers). Since the set is infinite, we cannot list all elements.
• Result: $A = \{1, 3, 5, 7, \dots\}$ (The ellipsis ... indicates the pattern continues infinitely).

(ii) $B = \{x : x \text{ is an integer, } -\frac{1}{2} < x < \frac{9}{2}\}$

• Domain: Integers ($\mathbb{Z}$). Integers are $\dots, -2, -1, 0, 1, 2, \dots$
• Condition: $x$ must be strictly greater than $-\frac{1}{2}$ AND strictly less than $\frac{9}{2}$.
  • $-\frac{1}{2} = -0.5$. Integers greater than $-0.5$ are $0, 1, 2, \dots$
  • $\frac{9}{2} = 4.5$. Integers less than $4.5$ are $\dots, 2, 3, 4$.
• Intersection: The integers that satisfy both are $0, 1, 2, 3, 4$.
• Result: $B = \{0, 1, 2, 3, 4\}$

(iii) $C = \{x : x \text{ is an integer, } x^2 \le 4\}$

• Domain: Integers ($\mathbb{Z}$).
• Condition: The square of $x$ must be less than or equal to $4$.
• Test candidates:
  • $x = -3 \implies (-3)^2 = 9 > 4$ (Reject)
  • $x = -2 \implies (-2)^2 = 4 \le 4$ (Keep)
  • $x = -1 \implies (-1)^2 = 1 \le 4$ (Keep)
  • $x = 0 \implies 0^2 = 0 \le 4$ (Keep)
  • $x = 1 \implies 1^2 = 1 \le 4$ (Keep)
  • $x = 2 \implies 2^2 = 4 \le 4$ (Keep)
  • $x = 3 \implies 3^2 = 9 > 4$ (Reject)
• Result: $C = \{-2, -1, 0, 1, 2\}$

(iv) $D = \{x : x \text{ is a letter in the word “LOYAL”}\}$ …

🔍 Mistake 1: Forgetting that “odd natural numbers” start from 1

Wrong answer:

$A = \{1, 3, 5, 7, 9, \dots\}$ is correct — but some students write $A = \{0, 1, 3, 5, \dots\}$ or $\{ \dots, -3, -1, 1, 3, \dots\}$.

Why it happens:

They confuse natural numbers (positive integers, usually starting at 1) with whole numbers or integers.

How to avoid:

• Memorise: In NCERT, natural numbers = $\{1, 2, 3, \dots\}$ (not 0).
• For “odd natural numbers”, the smallest is 1.

✓ Correct:

$A = \{1, 3, 5, 7, 9, \dots\}$

🔍 Mistake 2: Misreading the inequality in set B

Given:

$B = \{x : x \text{ is an integer}, \ -\frac{1}{2} < x < \frac{9}{2}\}$

Wrong answer:

Students include $-1$ or $0$ incorrectly, or include $4.5$ (not an integer).

Why it happens:

• They forget $-\frac{1}{2} < x$ means $x > -0.5$, so the smallest integer is $0$, not $-1$.
• They forget $\frac{9}{2} = 4.5$, so $x < 4.5$ means the largest integer is $4$, not $5$.

How to avoid:

• Convert fractions to decimals: $-\frac{1}{2} = -0.5$, $\frac{9}{2} = 4.5$.
• List integers strictly between: $0, 1, 2, 3, 4$.

✓ Correct:

$B = \{0, 1, 2, 3, 4\}$

🔍 Mistake 3: Forgetting negative integers in $x^2 \le 4$

Given:

$C = \{x : x \text{ is an integer}, \ x^2 \le 4\}$

Wrong answer:

$C = \{0, 1, 2\}$ (missing $-1$ and $-2$).

Why it happens:

Students think “square” means only positive numbers, or they solve $x \le 2$ instead of $|x| \le 2$.

How to avoid:

• Solve $x^2 \le 4$ as $|x| \le 2$, i.e., $-2 \le x \le 2$.
• List all integers in that range: $-2, -1, 0, 1, 2$.

✓ Correct:

$C = \{-2, -1, 0, 1, 2\}$

🔍 Mistake 4: Repeating letters in set D

Given:

$D = \{x : x \text{ is a letter in the word “LOYAL”}\}$

Wrong answer:

$D = \{L, O, Y, A, L\}$ (repeating L).

Why it happens:

They list letters as they appear, not as a set (which has no duplicates).

How to avoid:

• In a set, each element appears only once.
• Write unique letters: L, O, Y, A.

✓ Correct:

$D = \{L, O, Y, A\}$

🔍 Mistake 5: Including months with 31 days in set E

Given:

$E = \{x : x \text{ is a month of a year not having 31 days}\}$

Wrong answer:

$E = \{\text{February, April, June, September, November}\}$ — this is actually correct, but some students add January or March by mistake.

Why it happens:

They misread “not having 31 days” as “having 31 days”.

How to avoid:

• First list all months with 31 days: Jan, Mar, May, Jul, Aug, Oct, Dec.
• The complement (in a year) gives the answer.

✓ Correct:

$E = \{\text{February, April, June, September, November}\}$

--- …