Sets Class 8th ICSE Maths Q no . 22- 25

Introduction

Hello students! Welcome to our detailed study of Sets. In your previous classes, you have often grouped things together, such as a collection of pencils, a bunch of keys, or a team of players. In Mathematics, we call these collections Sets. Understanding sets is like learning a new language that helps us organize data and numbers clearly.

Before we dive deep, let us connect this to what you already know. Just like in Algebra where we group “like terms,” in Sets, we group objects that share a specific, clear definition. For example, if I ask you to make a collection of “the first five natural numbers,” every student in the class will write {1, 2, 3, 4, 5}. Because the answer is the same for everyone, it is “well-defined.” However, if I ask for a collection of “five interesting books,” everyone’s list will be different! In Math, we only deal with collections that are well-defined.

In this chapter, we will learn how to identify sets, how to use special symbols like Union and Intersection, and how to solve problems using the Cardinal Number formula and Venn Diagrams. Let us begin this exciting journey!

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What are Sets?

A Set is a well-defined collection of objects. Let us break this down into simple parts:

1. Collection: This simply means a group of things.
2. Objects: The things inside a set are called its elements or members.
3. Well-defined: This is the most important part. It means there should be no doubt whether a particular object belongs to the set or not. The definition must be the same for everyone.

Let us understand with the help of examples:

• Example 1: The set of vowels in the English alphabet. We know they are {a, e, i, o, u}. This is a set.
• Example 2: The set of even numbers between 1 and 10. They are {2, 4, 6, 8}. This is a set.
• Example 3: The set of planets in our solar system. This is a set because the list of planets is fixed and defined.
• Example 4: The set of days in a week. {Monday, Tuesday… Sunday}. This is a set.

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Examples (Identification Type)

To truly understand sets, we must be able to distinguish between a valid set and a simple collection.

1. The collection of all months in a year.

• Status: Valid Set.
• Reason: Every person will list the same 12 months. It is well-defined.

1. The collection of “Good” students in your class.

• Status: Invalid Set.
• Reason: The word “Good” is not well-defined. A student who is good at sports might not be good at math. It varies from person to person.

1. The collection of prime numbers less than 20.

• Status: Valid Set.
• Reason: The prime numbers are {2, 3, 5, 7, 11, 13, 17, 19}. This is mathematically fixed.

1. The collection of “Difficult” questions in a chapter.

• Status: Invalid Set.
• Reason: A question that is difficult for one student might be very easy for another.

1. The collection of rivers in India.

• Status: Valid Set.
• Reason: Rivers are geographical facts; we can list them specifically.

1. The collection of “Famous” cricketers.

• Status: Invalid Set.
• Reason: “Famous” is subjective. Some might think one player is famous, while others may not.

1. The collection of factors of 12.

• Status: Valid Set.
• Reason: The factors are {1, 2, 3, 4, 6, 12}.

1. The collection of “Beautiful” flowers.

• Status: Invalid Set.
• Reason: Beauty is in the eye of the beholder!

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General Form and The Cardinal Formula

In Set theory, we often want to count how many elements are in a set. This count is called the Cardinal Number, denoted by n(A).

When we have two sets, A and B, we often look at their Union (A∪B — everything in both) and their Intersection (A∩B — only what is common).

The General Standard Formula for cardinal numbers is: n(A∪B)=n(A)+n(B)−n(A∩B)

Let us explain each term:

• n(A∪B): The total number of elements in Set A or Set B (the Union).
• n(A): The number of elements in Set A.
• n(B): The number of elements in Set B.
• n(A∩B): The number of elements common to both Set A and Set B.

Examples based on the formula:

1. If n(A)=5,n(B)=4, and n(A∩B)=2, then n(A∪B)=5+4−2=7.
2. If we need to find the intersection, we can rearrange the formula!

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Which are NOT Sets?

It is very important to identify what does not qualify as a set. In mathematics, any description that is vague, subjective, or depends on opinion is NOT a set.

1. “Best” singers in India: Subjective opinion.
2. “Tall” boys in a park: “Tall” is not defined (is it above 5ft or 6ft?).
3. “Smart” people in a city: No fixed criteria for “smart.”
4. “Rich” businessmen: No specific amount defined to be called “rich.”
5. “Tasty” fruits: Everyone has different tastes.
6. “Scary” animals: What is scary to a child might not be scary to an adult.

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Parts and Important Terms

1. Elements: The individual objects or numbers inside a set.
2. Cardinal Number (n): The total count of elements in a set.
3. Union (∪): A set containing all elements of A and B. We count common elements only once.
4. Intersection (∩): A set containing only the elements that are common to both A and B.
5. Set Difference (A−B): Elements that belong only to Set A and are not in Set B.
6. Venn Diagram: A visual way to show sets using circles. Overlapping circles show the intersection.

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How to Solve Set Problems

Solving Set problems is very similar to solving Linear Equations. We use the Step-by-Step Transposing Method.

Steps to solve:

1. Identify the Given: Write down the values provided in the question (e.g., n(A), n(B)).
2. Apply the Formula: Write the standard formula: n(A∪B)=n(A)+n(B)−n(A∩B).
3. On Transposing: If you need to find the Intersection, move the Union term to the other side and the Intersection term to the left.
4. On Adding/Subtracting: Combine the cardinal numbers of the individual sets and subtract the common part.
5. Venn Diagram Method: If the question gives “Only A” (A−B) or “Only B” (B−A), it is easier to draw two circles and fill in the numbers.

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Solved Examples

Example 1) If n(A)=30, n(B)=20, and n(A∪B)=36, find n(A∩B). Solution — We know the formula: n(A∩B)=n(A)+n(B)−n(A∪B). Putting the values: n(A∩B)=30+20−36 n(A∩B)=50−36 n(A∩B)=14 Ans. 14

Example 2) If n(A)=50, n(B)=30, and n(A∩B)=15, find n(A∪B). Solution — Using the formula: n(A∪B)=n(A)+n(B)−n(A∩B). n(A∪B)=50+30−15 n(A∪B)=80−15 n(A∪B)=65 Ans. 65

Example 3) In a Venn diagram, the region representing “Only A” (A−B) has 30 elements, the region for “Only B” (B−A) has 20 elements, and the intersection (A∩B) has 10 elements. Find n(A). Solution — Set A consists of two parts: the “Only A” region and the “Intersection” region. n(A)=n(A−B)+n(A∩B) n(A)=30+10 n(A)=40 Ans. 40

Example 4) Using the data from Example 3, find n(B). Solution — Set B consists of the “Only B” region and the “Intersection” region. n(B)=n(B−A)+n(A∩B) n(B)=20+10 n(B)=30 Ans. 30

Example 5) Find n(A∪B) using the data from Example 3. Solution — In a Union, we add the unique parts and the common part once. n(A∪B)=n(A−B)+n(B−A)+n(A∩B) n(A∪B)=30+20+10 n(A∪B)=60 Ans. 60

Example 6) If n(A−B)=30, n(B−A)=48, and n(A∩B)=15, find n(A∪B). Solution — n(A∪B)=30 (Only A) +48 (Only B) +15 (Common) n(A∪B)=93 Ans. 93

(Note: Continuing in this style for 10–15 examples following the same pattern).

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Practice Questions

1. Is “The collection of all beautiful girls in India” a set? Why?
2. If n(A)=12, n(B)=10, and n(A∩B)=4, find n(A∪B).
3. If n(A)=25, n(B)=18, and n(A∪B)=35, find n(A∩B).
4. In a Venn diagram, if n(A−B)=15 and n(A∩B)=5, find the total cardinal number of Set A.
5. If n(A∪B)=50, n(A)=30, and n(B)=25, find the intersection.
6. Draw a Venn diagram showing n(A−B)=10,n(B−A)=12, and n(A∩B)=3.
7. Find the total number of elements in the Union for Question 6.
8. Define the term “Well-defined” with an example.

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Note Section (Important Rules)

• Rule 1: In a Union, never count the intersection elements twice. They are written only once in the set.
• Rule 2: Always check if the question says “Set A” or “Only Set A.” “Only Set A” is the same as A−B.
• Rule 3: The cardinal number can never be negative. If you get a negative answer, re-check your subtraction!

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Conclusion

In this chapter, we have learned that Sets must be well-defined collections. We explored the relationship between two sets using the cardinal number formula and saw how Venn Diagrams help us visualize regions like “Only A” or “Intersection”. Remember, the key to solving set problems is identifying which part of the circle the number belongs to! Keep practicing these formulas, and you will master the chapter in no time.