Q.Write the set {x : x is a positive integer and x2 < 40} in the roster form.

We need positive integers $x$ such that $x^2<40$.

Testing values: $6^2=36<40$, but $7^2=49$ is NOT less than $40$. So the largest qualifying integer is $6$, and every positive integer from $1$ to $6$ satisfies the condition.

Roster form means listing all the elements of a set inside curly braces, separated by commas -- no conditions, just the actual values.

The condition is: $x$ is a positive integer and $x^2 < 40$.

Step 1: Start from the smallest positive integer and test each one.

$1^2=1<40$ -- include $1$.

$2^2=4<40$ -- include $2$.

$3^2=9<40$ -- include $3$.

$4^2=16<40$ -- include $4$.

$5^2=25<40$ -- include $5$.

$6^2=36<40$ -- include $6$.

Step 2: Find the cutoff.

$7^2=49$, which is NOT less than $40$, so $7$ is excluded. Every integer larger than $7$ has an even bigger square, so none of them qualify either.

Step 3: Collect the qualifying integers into roster form.

Concept: Interval Notation & Roster Form

Step 1: Identify the condition

We need positive integers $x$ such that $x^2 < 40$.

Step 2: Find the range of $x$

Since $x$ is a positive integer, $x \geq 1$.

Solve $x^2 < 40 \implies x < \sqrt{40} \approx 6.324$.

So $x$ can be $1, 2, 3, 4, 5, 6$.

Step 3: Verify each integer

$1^2=1$, $2^2=4$, $3^2=9$, $4^2=16$, $5^2=25$, $6^2=36$ — all are $<40$.

$7^2=49 > 40$, so stop.

Final Answer:

$\{1, 2, 3, 4, 5, 6\}$

Common Mistakes in Writing Interval Notation

Mistake 1: Confusing Open and Closed Brackets

The Error:

Students often write (–4, 6] as [–4, 6] or (–4, 6), misreading the inequality signs.

Why It Happens:

• < and > mean open (excluded endpoint) → use ( or )
• ≤ and ≥ mean closed (included endpoint) → use [ or ]

How to Avoid:

Draw a number line for each inequality. Mark the endpoint with:

• An open circle (○) for < or >
• A filled circle (●) for ≤ or ≥

Then write the bracket that matches the circle.

Correct answers:

• (i) $(-4, 6]$
• (ii) $(-12, -10)$
• (iii) $[0, 7)$
• (iv) $[3, 4]$

Mistake 2: Reversing the Order of Endpoints

The Error:

Writing (6, –4] instead of (-4, 6] — putting the larger number first.

Why It Happens:

Students copy the order from the set-builder form without thinking about the number line.

How to Avoid:

Always write the smaller number on the left. On a number line, values increase left to right. Interval notation must follow this natural order.

Memory trick:

"Left is less, right is greater — always write small → big."

Mistake 3: Forgetting the Variable is Real Numbers

The Error:

Writing (-4, 6] but thinking it only includes integers like {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}.

Why It Happens:

Students confuse interval notation (continuous set of real numbers) with roster form (listing discrete elements).

How to Avoid:

Remember: $x \in \mathbb{R}$ means all real numbers between the endpoints — including fractions, decimals, and irrationals. The interval (-4, 6] contains infinitely many numbers, not just integers.

Mistake 4: Mixing Up Infinity Notation

The Error:

Writing [–∞, 6] or (–12, ∞] — using closed brackets with infinity.

Why It Happens:

Students treat infinity like a number that can be "reached" or "included."

How to Avoid:

Infinity ($\infty$) is not a number — it's a concept meaning "unbounded." You can never include it, so always use:

• (–∞, ... (open on left)
• ..., ∞) (open on right)

Correct examples:

• $(-\infty, 5]$ — includes 5, but not infinity
• $(3, \infty)$ — excludes 3, and infinity is always open

Quick Reference Table

Final tip: Practice by converting back and forth:

Set-builder → Number line → Interval notation.

This triple-check catches all four mistakes.