Q.Write the following sets in the set-builder form :

Concept: Set-Builder Form

Set-builder notation describes a set by stating the property that its elements satisfy, written as $\{x : \text{condition on } x\}$ or $\{x \mid \text{condition on } x\}$.

Solution:

(i) The elements are multiples of 3 up to 12. We write $\{x : x = 3n, \, n \in \mathbb{N}, \, 1 \le n \le 4\}$ or equivalently $\{3n : n \in \mathbb{N}, \, 1 \le n \le 4\}$.

(ii) Each term is a power of 2: $2^1, 2^2, 2^3, 2^4, 2^5$. So $\{2^n : n \in \mathbb{N}, \, 1 \le n \le 5\}$.

(iii) Each term is a power of 5: $5^1, 5^2, 5^3, 5^4$. So $\{5^n : n \in \mathbb{N}, \, 1 \le n \le 4\}$.

(iv) All positive even integers. We write $\{2n : n \in \mathbb{N}\}$ or $\{x : x \in \mathbb{N}, \, x \text{ is even}\}$. …

Understanding Set-Builder Form

When we list elements explicitly—like $\{3, 6, 9, 12\}$—we're using roster form. But what if the pattern continues indefinitely, or we want to capture the rule that generates the elements? That's where set-builder notation shines.

The structure is always: $\{x : \text{property that } x \text{ satisfies}\}$ or equivalently $\{x \mid \text{property}\}$. The colon (or vertical bar) reads as "such that." Our job is to spot the pattern and express it as a mathematical condition.

Solutions

(i) $\{3, 6, 9, 12\}$

1. Identify the pattern: Each element is a multiple of 3. We have $3 = 3 \times 1$, $6 = 3 \times 2$, $9 = 3 \times 3$, $12 = 3 \times 4$.

2. Determine the range: The multiplier runs from 1 to 4, so we need $x = 3n$ where $n \in \{1, 2, 3, 4\}$.

3. Write in set-builder form:

Alternatively, since these are the first four positive multiples of 3:

(ii) $\{2, 4, 8, 16, 32\}$

1. Spot the exponential pattern: Each term is double the previous one. Writing them as powers: $2 = 2^1$, $4 = 2^2$, $8 = 2^3$, $16 = 2^4$, $32 = 2^5$.

2. Express the general term: The elements are $x = 2^n$ where the exponent $n$ ranges from 1 to 5.

3. Set-builder form:

(iii) $\{5, 25, 125, 625\}$

1. Recognize powers of 5: We have $5 = 5^1$, $25 = 5^2$, $125 = 5^3$, $625 = 5^4$.

2. Formulate the condition: Elements are $x = 5^n$ with $n$ from 1 to 4.

3. Set-builder form:

(iv) $\{2, 4, 6, \ldots\}$

1. Identify the infinite pattern: The ellipsis indicates the set continues indefinitely. These are all positive even integers.

2. Express as multiples: An integer is even if it equals $2n$ for some natural number $n$. Since we want all positive evens, $n$ starts at 1.

3. Set-builder form:

   $\{x : x = 2n, \, n \in \mathbb{N}\}$ …

Set Membership: Writing Sets in Set-Builder Form

Method: Pattern Recognition & Set-Builder Notation

The core idea is to identify the pattern in the given elements, then express that pattern using a variable (usually $x$) and a condition.

Steps for any such problem:

1. Observe the elements — look for a common rule (multiplication, addition, powers, etc.)
2. Identify the domain — what type of numbers are these? (natural numbers, integers, etc.)
3. Write the general form — express a typical element using a variable
4. State the condition — what values can the variable take?

(i) $\{3, 6, 9, 12\}$

Pattern: Multiples of 3, from $3 \times 1$ to $3 \times 4$

Set-builder form:

(ii) $\{2, 4, 8, 16, 32\}$

Pattern: Powers of 2: $2^1, 2^2, 2^3, 2^4, 2^5$

Set-builder form:

(iii) $\{5, 25, 125, 625\}$

Pattern: Powers of 5: $5^1, 5^2, 5^3, 5^4$

Set-builder form:

(iv) $\{2, 4, 6, \dots\}$ …

(i) $\{3, 6, 9, 12\}$

Common Mistakes

• Writing the rule too loosely

  Example: $\{ x : x \text{ is a multiple of 3} \}$

  This is wrong because it would also include $15, 18, \dots$ — not just the four given numbers.

• Forgetting to restrict the domain

  Students often forget to say “$x$ is a natural number” or “$x \in \mathbb{N}$”.

How to Avoid

• Always check: does my rule produce exactly the listed elements and no extras?
• Use a stopping condition like $x \leq 12$ or $x \in \{1,2,3,4\}$.

Correct form:

(ii) $\{2, 4, 8, 16, 32\}$

Common Mistakes

• Writing $\{ x : x = 2^n, n \in \mathbb{N} \}$

  This includes $2^0 = 1$ (if $0 \in \mathbb{N}$) and $2^6 = 64$, etc. — too many elements.

• Confusing powers of 2 with multiples of 2

  Some write $\{ x : x = 2n \}$ — that gives $\{2,4,6,8,\dots\}$, which is wrong.

How to Avoid

• Clearly state the exponent range: $n = 1,2,3,4,5$.
• Double-check: does $2^1$ give the first element? Yes.

Correct form:

(iii) $\{5, 25, 125, 625\}$

Common Mistakes

• Writing $\{ x : x = 5^n, n \in \mathbb{N} \}$

  Again, too broad — includes $5^0 = 1$, $5^5 = 3125$, etc.

• Mistaking the pattern

  Some think it’s multiples of 5: $\{ x : x = 5n \}$ — that gives $5,10,15,\dots$, not powers.

How to Avoid

• Recognise the pattern: each term is $5$ raised to $1,2,3,4$.
• Restrict $n$ to $1 \leq n \leq 4$.

Correct form:

(iv) $\{2, 4, 6, \dots\}$

Common Mistakes

• Writing $\{ x : x = 2n, n \in \mathbb{N} \}$ without clarifying if $0 \in \mathbb{N}$

  If $\mathbb{N}$ includes $0$, then $0$ is also in the set — but the given set starts at $2$.

• Forgetting to say “even natural numbers”

  Some write $\{ x : x \text{ is even} \}$ — this includes negative evens and zero, which are not in the list.

How to Avoid

• Always specify the universe (e.g., natural numbers).
• If the pattern is infinite, use “…” in set-builder or say “$n \in \mathbb{N}$” and ensure the first term matches.

Correct form:

(assuming $\mathbb{N} = \{1,2,3,\dots\}$)

(v) $\{1, 4, 9, \dots, 100\}$

Common Mistakes

• Writing $\{ x : x = n^2, n \in \mathbb{N} \}$

  This includes $121, 144, \dots$ — no upper bound.

• Forgetting the upper limit …