Solutions of Unit 17 Sets Class 10 Sindh Board

💡 Introduction: Why This Unit Is So Important

Unit 17 on Sets is one of the most important chapters in Class 10 Mathematics for Sindh Board students. This unit introduces you to a new “language” of mathematics – the language of sets.

You will see sets again and again in:

• Probability
• Data handling
• Algebra
• Computer science
• Logic and reasoning

Many students find this unit confusing, not because it is very difficult, but because they try to memorize instead of understanding the ideas. In this guide, we will study Sets in a simple, step-by-step way, just like a teacher explains in class.

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📘 What You Will Learn in Unit 17 sets class 10 Sindh board

1. Number Sets and Symbols (N, W, Z, E, O, P, C, Q, R)

In this unit, you will meet different sets of numbers, written with special symbols:

• N – Natural numbers (1, 2, 3, …)
• W – Whole numbers (0, 1, 2, 3, …)
• Z – Integers (…, −2, −1, 0, 1, 2, …)
• E – Even numbers (…, −4, −2, 0, 2, 4, …)
• O – Odd numbers (…, −3, −1, 1, 3, …)
• P – Prime numbers (2, 3, 5, 7, …)
• C – Composite numbers (4, 6, 8, 9, 10, …)
• Q – Rational numbers (fractions, integers, terminating & repeating decimals)
• R – Real numbers (all rational + irrational numbers)

You will also learn how these sets are related – for example, every natural number is also a whole number, every whole number is an integer, and so on.

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2. Set Notation and Representation

A set is a collection of well-defined objects (numbers, letters, people, etc.).

You learn two main ways to write sets:

(i) Tabular / Roster Form

You list all elements inside curly braces { }:

A = {2, 4, 6, 8}

(ii) Set-Builder Form

You describe the elements using a rule:

A = {x | x is an even number less than 10}

Set-builder form has two parts:

1. What kind of elements (nature) – “x is an even number”
2. Their range – “less than 10”

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3. Types of Sets

You will study several important types of sets:

• Empty (Null/Void) Set (∅)
  A set with no elements.
  Example: {x | x is a natural number less than 0}
• Finite Set
  A set with a limited number of elements.
• Infinite Set
  A set with endlessly many elements (e.g. natural numbers).
• Singleton Set
  A set with exactly one element.
  Example: {7}
• Equal Sets
  Two sets with exactly the same elements.
  {2, 4, 6} = {6, 4, 2}
• Equivalent Sets
  Sets with the same number of elements, even if elements are different.

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4. Subsets, Proper Subsets and Power Sets

A subset is a set whose every element is inside another set.

• A ⊆ B means “A is a subset of B”.
• Proper subset: A ⊂ B means A is part of B but not equal to B.
• Every set is an improper subset of itself (A ⊆ A).

You also learn about power sets:

• Power set of A (written P(A)) is the set of all subsets of A.
• If a set has n elements, then number of subsets = 2ⁿ.

Example:
If A = {a, b}, then

P(A) = {∅, {a}, {b}, {a, b}}

Here n = 2 and 2² = 4 subsets.

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5. Operations on Sets

This is the most practical and exam-heavy part of the unit.

(i) Union (A ∪ B)

Union means “combine” elements of both sets, without repeating.

A = {1, 2}, B = {2, 3}
A ∪ B = {1, 2, 3}

(ii) Intersection (A ∩ B)

Intersection means “common” elements.

A ∩ B = {2}

(iii) Difference (A − B)

Difference means “elements in A that are not in B”.

A − B = {1}

Remember:
A − B ≠ B − A in general. Set difference is not commutative.

(iv) Complement (A′)

Complement means “elements that are in the universal set, but not in A”.

If U = {1, 2, 3, 4, 5} and A = {2, 5}, then

A′ = {1, 3, 4}

You will also see important identities like:

• A − B = A ∩ B′
• A − ∅ = A
• A − A = ∅

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6. Special Relationships Between Sets

You learn to classify pairs of sets:

Disjoint Sets

Two sets are disjoint if they have no element in common.

A ∩ B = ∅

Example:
A = {1, 2}, B = {3, 4}

Overlapping Sets

Two sets are overlapping if they have at least one common element, and neither is a subset of the other.

Example:
A = {1, 2, 3}, B = {3, 4, 5}

Common element = {3}

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7. Venn Diagrams

Venn diagrams help you see the sets and their relations:

• Rectangle = Universal set (U)
• Circles = Sets (A, B, C…)
• Overlap = Intersection
• Combined area = Union
• Outside circle but inside rectangle = Complement

Many questions in this unit use shaded regions in Venn diagrams.
If you learn to draw and read them carefully, half the chapter becomes easy.

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8. Properties of Set Operations

This unit also teaches which operations are:

• Commutative (order doesn’t matter)
  • A ∪ B = B ∪ A
  • A ∩ B = B ∩ A
  • But A − B ≠ B − A
• Associative (grouping doesn’t matter)
  • (A ∪ B) ∪ C = A ∪ (B ∪ C)
  • (A ∩ B) ∩ C = A ∩ (B ∩ C)
• Distributive
  • A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
  • A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Understanding these properties helps in proving identities and solving MCQs.

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⚠️ Common Mistakes Students Make (And How to Fix Them)

Let’s combine the most frequent and dangerous mistakes:

❌ Mistake 1: Confusing ∈ with ⊆

• Wrong: {2} ∈ {1, 2, 3}
• Right: {2} ⊆ {1, 2, 3} and 2 ∈ {1, 2, 3}

How to remember:
Use ∈ when talking about a single element and a set.
Use ⊆ when comparing one set with another set.

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❌ Mistake 2: Thinking All Operations Are Commutative

Many students think A − B = B − A.
This is wrong.

Example:
A = {1, 2, 3}, B = {3, 4, 5}

• A − B = {1, 2}
• B − A = {4, 5}

They are not equal.

Union and intersection are commutative, but difference is not.

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❌ Mistake 3: Misunderstanding the Empty Set (∅)

• ∅ is a set with no elements.
• {∅} is a set that has one element, and that element is the empty set.

Also, the empty set does have subsets. In fact:

• Power set of ∅ is {∅}.
• It is not equal to ∅.

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❌ Mistake 4: Wrong Venn Diagrams and Shading

Students often:

• Show overlap where sets are disjoint
• Shade the wrong region in A − B, (A ∪ B)′, etc.

Fix:
Always start with a small example using numbers, then draw the diagram. Think:

• Union = total area of A and B
• Intersection = only overlapping area
• Difference A − B = part of A without B
• Complement A′ = area outside A

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❌ Mistake 5: Messy or Wrong Set-Builder Notation

Students sometimes forget to write the “condition” properly.

Correct structure is:

{x | x has this property}

Example:

A = {x | x is an even natural number less than 20}

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❌ Mistake 6: Miscalculating Number of Subsets

Students forget to include ∅ or the set itself.

Always use:

Number of subsets of a set with n elements = 2ⁿ

Example:
If A has 3 elements, subsets = 2³ = 8 (including ∅ and A itself).

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❌ Mistake 7: Not Recognizing A − B = A ∩ B′

Some students don’t notice that:

A − B and A ∩ B′ are the same thing.

Once you understand this, many problems become easier — especially with Venn diagrams.

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🧑‍🏫 How to Prepare This Unit – Teacher’s Strategy

Here is a simple “study plan” based on how an experienced teacher would guide you.

🔹 Phase 1: Build Strong Foundations (Week 1)

• Learn all symbols: ∈, ∉, ⊆, ⊂, ∪, ∩, −, ′, ∅, U.
• Practice writing sets in both tabular and set-builder form.
• Start with examples from the textbook and convert them from one form to another.
• Use real objects (pens, coins, colours) to form sets and understand the idea.

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🔹 Phase 2: Operations and Venn Diagrams (Week 2)

• Practice union and intersection using small sets.
• Learn difference and complement with Venn diagrams.
• For every question with words like “only”, “at least”, “neither”, “both”, draw a diagram first.
• Work carefully through the solved examples in the textbook and your notes.

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🔹 Phase 3: Properties and Identities (Week 3)

• Make a table of which operations are commutative, associative, and distributive.
• Practice questions that involve proving set identities.
• Try to start from the more complicated side of the identity and reach the other side step by step.

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🔹 Phase 4: Full Practice & Revision (Week 4)

• Solve all textbook exercises of Unit 17.
• Then solve past paper questions from this unit.
• Check your mistakes and mark topics where you are weak.
• Revisit those topics using your notes and this guide.

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⏰ Sample 4-Week Preparation Timeline

Week 1 – Basics

• Types of sets, number sets (N, W, Z, E, O, P, C, Q, R)
• Set notation and representation
• Subsets and power sets

Week 2 – Operations

• Union and intersection
• Difference and complement
• Venn diagrams (2 sets)

Week 3 – Properties & Relations

• Disjoint and overlapping sets
• Commutative, associative, distributive properties
• Mixed operation questions

Week 4 – Exam Focus

• All textbook exercises
• Past paper questions
• Full revision and mistake correction

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📝 Expert Exam Tips for Unit 17

1. Time Management
   Don’t spend 10 minutes on a 2-mark question. Match your time with the marks.
2. Show All Steps
   Always show working, especially in set identities and Venn diagram questions.
3. Double-Check Answers
   Use a second method when possible – list elements, use diagrams, or apply formulas.
4. Memorize Key Formulas Clearly
   • Number of subsets = 2ⁿ
   • n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
   • A − B = A ∩ B′
   • De Morgan’s Laws:
     • (A ∪ B)′ = A′ ∩ B′
     • (A ∩ B)′ = A′ ∪ B′
5. Practice 3-Set Venn Diagrams
   These can be tricky but are common in exams. Learn to handle each of the 8 regions step by step.

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🎯 How to Use This Page and the Solved Exercises

On this page, you will find:

• This complete explanation of Unit 17: Sets
• Solved exercises of this unit (links added by you below)

For Best Results:

1. First, read this guide slowly and understand the ideas.
2. Then, open the solved exercises for each section.
3. Try each question yourself first, then compare with the solution.
4. Mark questions you find difficult and repeat them later.

This way, you are not just copying answers, but really learning the chapter.

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⚖️ Educational Disclaimer

All the material on this page is prepared only for educational purposes to help Class 10 (Sindh Board) students understand their Mathematics syllabus more easily.

• Explanations, summaries, and solved examples are created independently to support learning and do not replace the official Sindh Textbook Board book.
• Any reference to the official textbook, its exercises, or its notation is made strictly for academic guidance.
• This website does not claim any ownership of the Sindh Textbook Board textbook and fully respects all related rights.

Our aim is to provide fair-use, student-friendly educational support so that students can understand the subject better, improve their concepts, and perform well in exams—without any commercial misuse or unauthorized copying of protected material.

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🌟 Final Words of Encouragement

Set Theory may look a little abstract at first, but it is one of the most logical and scoring units in Class 10 Mathematics. Once you understand the basic ideas of sets, subsets, and operations, this unit becomes easy and even fun.

If you:

• Read this guide carefully
• Practice all the solved exercises
• Revise your mistakes honestly

then Unit 17 can become one of your strongest chapters in the exam and a powerful base for F.Sc., engineering, computer science, and many other fields.

You can do it — one set at a time.

You can download the pdf books of class 10 maths textbooks for sindh and punjab boards.