Comprehensive math worksheet designed to help students practice converting sets to set-builder form and identifying valid sets.

Let's solve the problems step by step from the worksheet.

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Worksheet on Sets in Set-builder Form

#### 1. Write each given set in the Set-Builder Form:

We need to express each set using a rule or property that defines its elements.

(i) {2, 4, 6, 8, 10}
These are even numbers from 2 to 10.
Set-builder form:
$$
\{x \mid x \text{ is an even natural number and } 2 \leq x \leq 10\}
$$

(ii) {2, 3, 5, 7, 11}
These are prime numbers less than 12.
Set-builder form:
$$
\{x \mid x \text{ is a prime number and } x < 12\}
$$

(iii) {January, June, July}
These are months of the year with names starting with 'J'.
Set-builder form:
$$
\{x \mid x \text{ is a month of the year and } x \text{ starts with the letter J}\}
$$

(iv) {a, e, i, o, u}
These are vowels in the English alphabet.
Set-builder form:
$$
\{x \mid x \text{ is a vowel in the English alphabet}\}
$$

(v) {Tuesday, Thursday}
These are days of the week that are not weekends and have "day" in their name (but better to use a clearer rule).
Alternatively: Days of the week that are midweek and not Monday, Wednesday, Friday, etc.
Better:
$$
\{x \mid x \text{ is a day of the week and } x \text{ is neither a weekend nor a weekday ending in 'day'}\}
$$
But this is messy. A simpler and correct way:
$$
\{x \mid x \text{ is a day of the week and } x \in \{\text{Tuesday, Thursday}\}\}
$$
But we want a *rule*, so perhaps:
$$
\{x \mid x \text{ is a day of the week and } x \text{ has exactly 8 letters}\}
$$
Wait — Tuesday = 7, Thursday = 8 → no.
Actually, both are weekdays that are not adjacent to weekends? Not helpful.

Best approach: Since it's just two specific days, but we must use a rule:
Notice: These are the only days of the week whose names contain the letter 'u'?
Tuesday → yes, Thursday → yes.
Monday → no, Wednesday → no, Friday → no, Saturday → no, Sunday → no.
So:
$$
\{x \mid x \text{ is a day of the week and } x \text{ contains the letter 'u'}\}
$$
Yes! That works.

So:
$$
\{x \mid x \text{ is a day of the week and } x \text{ contains the letter 'u'}\}
$$

(vi) {1, 4, 9, 16, 25}
These are perfect squares: $1^2, 2^2, 3^2, 4^2, 5^2$
Set-builder form:
$$
\{x \mid x = n^2, n \in \mathbb{N}, 1 \leq n \leq 5\}
$$

(vii) {5, 10, 15, 20, 25, 30}
These are multiples of 5 from 5 to 30.
Set-builder form:
$$
\{x \mid x \text{ is a multiple of } 5 \text{ and } 5 \leq x \leq 30\}
$$

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#### 2. Write the following sets in Set-Builder Form or Rule form:

(i) A = {1, 3, 5, 7, 9}
Odd numbers from 1 to 9.
Set-builder form:
$$
\{x \mid x \text{ is an odd natural number and } 1 \leq x \leq 9\}
$$

(ii) B = {16, 25, 36, 49, 64}
Perfect squares: $4^2, 5^2, 6^2, 7^2, 8^2$
Set-builder form:
$$
\{x \mid x = n^2, n \in \mathbb{N}, 4 \leq n \leq 8\}
$$

(iii) C = {a, e, i, o, u}
Same as (iv) above.
Set-builder form:
$$
\{x \mid x \text{ is a vowel in the English alphabet}\}
$$

(iv) D = {violet, indigo, blue, green, yellow, orange, red}
These are colors of the rainbow (ROYGBIV).
Set-builder form:
$$
\{x \mid x \text{ is a color of the rainbow}\}
$$

(v) E = {January, March, May, July, August, October, December}
These are months with 31 days.
Set-builder form:
$$
\{x \mid x \text{ is a month of the year and } x \text{ has 31 days}\}
$$

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Worksheet on Elements Form a Set

Now determine whether the following collections form a set or not. A set must have well-defined elements — meaning it should be possible to determine whether any given object belongs to the collection or not.

#### 1. All easy problems in your text book.
→ Not a set, because "easy" is subjective. What is easy for one person may not be easy for another.
Reason: The term "easy" is not well-defined.

#### 2. Capitals of all countries in Asia.
→ Yes, it forms a set.
Each country has a capital, and there are a finite number of Asian countries. The capitals are clearly defined.
Answer: Yes.

#### 3. All three sided figures.
→ Yes, it forms a set.
A three-sided figure is a triangle. All triangles are well-defined geometric shapes.
Answer: Yes.

#### 4. First five counting numbers.
→ Yes, it forms a set.
Counting numbers: 1, 2, 3, 4, 5. Clearly defined.
Answer: Yes.

#### 5. All the tall boys of your class.
→ Not a set.
"Tall" is vague and subjective. There's no clear criterion to determine who is "tall".
Reason: The term "tall" is not well-defined.

#### 6. Last three days of the week.
→ Yes, it forms a set.
Assuming the week starts on Sunday: Friday, Saturday, Sunday.
Or if Monday starts: Friday, Saturday, Sunday — same.
The last three days are unambiguous: Friday, Saturday, Sunday.
Answer: Yes.

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✔ Final Answers:

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#### Part 1: Set-Builder Form

(i) $\{x \mid x \text{ is an even natural number and } 2 \leq x \leq 10\}$
(ii) $\{x \mid x \text{ is a prime number and } x < 12\}$
(iii) $\{x \mid x \text{ is a month of the year and } x \text{ starts with 'J'}\}$
(iv) $\{x \mid x \text{ is a vowel in the English alphabet}\}$
(v) $\{x \mid x \text{ is a day of the week and } x \text{ contains the letter 'u'}\}$
(vi) $\{x \mid x = n^2, n \in \mathbb{N}, 1 \leq n \leq 5\}$
(vii) $\{x \mid x \text{ is a multiple of } 5 \text{ and } 5 \leq x \leq 30\}$

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#### Part 2: Write in Set-Builder Form

(i) $\{x \mid x \text{ is an odd natural number and } 1 \leq x \leq 9\}$
(ii) $\{x \mid x = n^2, n \in \mathbb{N}, 4 \leq n \leq 8\}$
(iii) $\{x \mid x \text{ is a vowel in the English alphabet}\}$
(iv) $\{x \mid x \text{ is a color of the rainbow}\}$
(v) $\{x \mid x \text{ is a month of the year and } x \text{ has 31 days}\}$

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#### Part 3: Do these form a set?

1. No – "easy" is subjective.
2. Yes – Well-defined capitals.
3. Yes – Triangles are well-defined.
4. Yes – First five counting numbers: 1,2,3,4,5.
5. No – "tall" is subjective.
6. Yes – Last three days: Friday, Saturday, Sunday.

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