Exercise 3.1 is the first step into a brand-new chapter — Sets and Functions. Chapter 2 was all about Logarithms. Now you start with the simplest idea in the whole chapter: what a set is, and the three ways to write one. The whole exercise runs on three forms — Descriptive form, Tabular form and Set builder notation — plus two new ideas, Subset and Power set. These notes cover Exercise 3.1 completely, in handwritten style and in very simple words. They are written for slow learners and for students whose first language is not English, so every step is short, every new word is explained the moment it appears, and every answer is checked.
Read This First: The 3 Ways to Write a Set
A set is a group of clear, different things. The things in a set are called its elements, and we write a set inside curly brackets: { }. There are three ways to write the same set, and this exercise makes you practise all three. Descriptive form — say it in words (“the set of vowels”). Tabular form — list every element ({a, e, i, o, u}). Set builder notation — give the rule ({x | x is a vowel}). Learn the three forms first, and every question in this exercise is just moving between them. Two more ideas come with them: a Subset is a smaller set living inside a bigger one, and a Power set is the set of ALL the subsets — if a set has n elements, its power set has 2ⁿ elements.
Notes Details
| Class | Class 9 / Grade 9 |
| Subject | Mathematics |
| Chapter | 3 – Sets and Functions |
| Exercise | 3.1 (complete) |
| Format | PDF (handwritten notes) |
| Pages | 16 |
| Language | Simple English |
| Price | Free |
For International Students
These notes are labelled “Class 9”, the name used in Pakistan, India and Bangladesh — but the mathematics is the same everywhere. Class 9 in Pakistan is Grade 9 in the United States and Canada, and Year 10 in the United Kingdom, where sets appear in GCSE number and algebra topics. Students in the Gulf following a Pakistani, Indian or British curriculum school use the same chapter. The English here is deliberately simple: short sentences, everyday words, and no heavy textbook language, so the notes work just as well for a student whose first language is not English. They follow the PCTB (Punjab Curriculum and Textbook Board) Class 9 Mathematics book, Chapter 3 — Sets and Functions, and they are equally useful for other BISE boards, and for CBSE, ICSE and NCTB students studying the same sets topics. You can read the whole exercise on this page and download the free PDF — no sign-up, no payment, no email address.
What Is Included in These Notes?
1. The Toolbox (Pages 2–3)
What a set is, the three forms — Descriptive, Tabular and Set builder notation — the number sets (N, W, Z, Q, R, P for primes), and Subset and Power set explained in plain words.
2. Question 1 — Set Builder Notation (Pages 4–6)
All nine parts: turn a list into a rule. Answers: {x | x = n², 1 ≤ n ≤ 22}, {x | x = 2ⁿ, 1 ≤ n ≤ 8}, {x | x ∈ Z, −1000 ≤ x ≤ 1000}, {x | x = 6n, 1 ≤ n ≤ 20}, {x | x = 100 + 2n, n ∈ W, 0 ≤ n ≤ 150}, {x | x = 3ⁿ, n ∈ W}, {x | x is a divisor of 100}, {x | x = 5n, 1 ≤ n ≤ 20}, and {x | x ∈ Z, −100 < x < 1000}.
3. Question 2 — Tabular Form (Pages 7–8)
All eight parts: read the rule, then list the elements. Answers: {3, 6, 9, …, 33}, {−1/2}, {2, 3, 5, 7, 11}, {1, 2, 4, 8, 16, 32, 64, 128}, {2, 4, 8, 16, 32, 64, 128}, { } (the empty set), {1, 2, 3, 4, …}, and { } (the empty set again).
4. Question 3 — Two Proper Subsets (Pages 9–10)
All seven parts: a proper subset must miss at least one element. Answers include {a} and {b} for {a, b, c}, the primes and the even naturals inside N, and {1/2} and {3/4} for the rationals between 0 and 2.
5. Question 4 — A Set With No Proper Subset (Page 10)
One question, one answer: yes — the empty set { } has no proper subset, because nothing can be smaller than nothing.
6. Question 5 — Spot the Difference (Page 11)
{a, b} has 2 elements. {{a, b}} has 1 element — the set {a, b} itself, like a box inside a box.
7. Question 6 — How Big is the Power Set? (Pages 11–12)
All six parts, using the rule: number of elements of P(A) = 2ⁿ. Answers: 1, 4, 128, 256, 4 and 8.
8. Question 7 — Write the Power Set (Pages 13–14)
All four parts, listing every subset in order. Answers: P({9, 11}) = {∅, {9}, {11}, {9, 11}}, all 16 subsets of {+, −, ×, ÷}, P({∅}) = {∅, {∅}}, and P({a, {b, c}}) = {∅, {a}, {{b, c}}, {a, {b, c}}}.
9. Check Your Answers (Pages 15–16)
Every answer from all seven questions in one table, so you can check your own work in a minute.
Exercise 3.1 Detailed Walkthrough: Every Question Explained
Question 1 — Set builder notation.
The trick is always the same three questions: What is the rule? Where does it start? Where does it stop? For (i), {1, 4, 9, 16, 25, 36, …, 484} — every number is a square, x = n², and 484 = 22², so n goes from 1 to 22. For (ii), {2, 4, 8, 16, …, 256} — powers of 2, x = 2ⁿ, and 256 = 2⁸. For (iii), {0, ±1, ±2, …, ±1000} — all the integers from −1000 to 1000, so {x | x ∈ Z, −1000 ≤ x ≤ 1000}. For (iv), multiples of 6 up to 120: x = 6n, n from 1 to 20. For (v), {100, 102, 104, …, 400} — start at 100 and add 2n, so x = 100 + 2n, and n starts at 0, so n is in W, not N. For (vi), {1, 3, 9, 27, 81, …} — powers of 3 that never stop: {x | x = 3ⁿ, n ∈ W}. For (vii), {1, 2, 4, 5, 10, 20, 25, 50, 100} — every one divides 100 exactly, so they are the divisors of 100. For (viii), multiples of 5 up to 100: x = 5n, n from 1 to 20. For (ix), the integers between −100 and 1000 — “between” means the end numbers are not counted, so use <, not ≤: {x | x ∈ Z, −100 < x < 1000}. The pattern trick to remember: always ask three things — the rule, the start, the stop.
Question 2 — Tabular form.
Now go the other way: read the rule, then list the elements. For (i), multiples of 3 up to 35: 3, 6, 9, …, 33 — stop, because 36 is more than 35. For (ii), {x | x ∈ R, 2x + 1 = 0} — solve it, x = −1/2, and −1/2 is a real number, so the answer is {−1/2}. For (iii), primes less than 12: {2, 3, 5, 7, 11}. For (iv), divisors of 128: {1, 2, 4, 8, 16, 32, 64, 128} — each one is double the one before. For (v), x = 2ⁿ with n < 8: {2, 4, 8, 16, 32, 64, 128}. For (vi), {x | x ∈ N, x + 4 = 0} — x = −4, but −4 is not a natural number, so nothing fits: the empty set { }. For (vii), {x | x ∈ N, x = x} — every number equals itself, so every natural number fits: {1, 2, 3, 4, …}. For (viii), {x | x ∈ Z, 3x + 1 = 0} — x = −1/3, but −1/3 is not an integer, so again the empty set { }. The big trap in this question: solving the equation is only half the work — always check that the answer belongs to the set named in the rule.
Question 3 — Two proper subsets.
A proper subset is smaller — it must miss at least one element. Many answers are possible; these are good ones. For (i) {a, b, c}: {a} and {b}. For (ii) {0, 1}: {0} and {1}. For (iii) N: the set of prime numbers and the set of even natural numbers — both live inside N, but neither is all of N. For (iv) Z: the primes and the whole numbers. For (v) Q: the integers and the natural numbers — every integer can be written as a fraction, like 3 = 3/1. For (vi) R: the rationals and the irrationals. For (vii) {x | x ∈ Q, 0 < x ≤ 2}: pick two simple fractions that fit — {1/2} and {3/4}.
Question 4 — A set with no proper subset.
Is there any set which has no proper subset? Yes. The empty set { } has no elements at all, and nothing can be smaller than nothing — so it has no proper subset.
Question 5 — Spot the difference.
What is the difference between {a, b} and {{a, b}}? Count the elements — that is the whole trick. {a, b} has two elements, a and b, separate and side by side. {{a, b}} has only one element: the set {a, b} itself, packed inside like a box inside a box.
Question 6 — How big is the power set?
Use the rule: number of elements of P(A) = 2ⁿ, where n is the number of elements in the set. For (i) { }: n = 0, so 2⁰ = 1 element. For (ii) {0, 1}: n = 2, so 2² = 4. For (iii) {1, 2, 3, 4, 5, 6, 7}: n = 7, so 2⁷ = 128. For (iv) {0, 1, 2, 3, 4, 5, 6, 7}: count carefully — that is n = 8, not 7, zero counts too — so 2⁸ = 256. For (v) {a, {b, c}}: a is one element and {b, c} is the second, so n = 2 and 2² = 4. For (vi) {{a, b}, {b, c}, {d, e}}: each small set is one element, so n = 3 and 2³ = 8. The trap in (v) and (vi): do not count the letters inside the small sets — count only the things separated by commas in the outer set. Learn the doubling table by heart: 2⁰ = 1, 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32, 2⁶ = 64, 2⁷ = 128, 2⁸ = 256.
Question 7 — Write the power set.
Now write all the subsets, not just count them. Work in order: first the empty set, then one element at a time, then two together, then the whole set. For (i) {9, 11}: n = 2, so P(A) must have 2² = 4 subsets — {∅, {9}, {11}, {9, 11}}. For (ii) {+, −, ×, ÷}: n = 4, so 2⁴ = 16 subsets — none (∅), one at a time (4), two at a time (6), three at a time (4), all four (1): 1 + 4 + 6 + 4 + 1 = 16, none missing. For (iii) {∅}: careful — this set is not empty, it has one element, the empty set itself — so n = 1 and P(A) = {∅, {∅}}. For (iv) {a, {b, c}}: n = 2, so 4 subsets — {∅, {a}, {{b, c}}, {a, {b, c}}} — note the double brackets. And the hardest idea in this exercise: ∅ means the empty set, 0 elements; {∅} means a set holding the empty set, 1 element. An empty bag is still one bag. Always check your power set: count your subsets when you finish — the count must be 2ⁿ. If it is not, one subset is missing.
Frequently Asked Questions
What are the 3 ways to write a set?
Descriptive form (words), Tabular form (list every element) and Set builder notation (give the rule). This exercise moves between all three, so learn them together, not one by one.
What is the difference between a subset and a proper subset?
A subset can be the whole set itself. A proper subset must be smaller — it must miss at least one element. That is why the empty set has no proper subset: nothing is smaller than nothing.
What is a power set, in simple words?
The set of ALL the subsets of a set. If a set has n elements, its power set has 2ⁿ elements. Count n with care — a set inside a set counts as one element.
What is the difference between ∅ and {∅}?
∅ is the empty set: 0 elements. {∅} is a set holding the empty set: 1 element. An empty bag is still one bag.
When is a set empty?
When nothing fits the rule — for example {x | x ∈ N, x + 4 = 0}, because x = −4 is not a natural number. Always check the answer belongs to the set named in the rule.
Which syllabus do these notes follow?
They follow the PCTB (Punjab Curriculum and Textbook Board) Class 9 Mathematics book, Chapter 3 — Sets and Functions, and they are equally useful for other BISE boards teaching the same chapter, and for international students studying sets at Grade 9 / Year 10 level.
Can I print these notes or share them with my class fellows?
Yes. The PDF prints cleanly on A4 paper, and you may share it freely with your class fellows and study groups.
Conclusion
Exercise 3.1 gives you the language of sets: three ways to write a set, what a subset is, and the power set with its 2ⁿ rule — then seven questions that make you use all of it, including the two classic traps (check the answer belongs to the named set, and count a set inside a set as one element). Download the free PDF, practise until “find the pattern first, then write the rule” is automatic, and Chapter 3 will feel started, not just opened. For the Logarithms chapter that came before, see Exercise 2.4, and find everything on our Free Class 9 Notes and Free Math Notes pages — everything on AllFreeAssets.com is free, always.
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