Sets Class 8th

Introduction

Welcome, students! Today, we are going to embark on an exciting journey into the world of Sets. Before we dive into the complexities of Chapter 6 from your Selina textbook, let us first connect this topic with something you already know from your previous classes. Think about how you organize your school bag. You have a pocket for books, a case for your pens, and perhaps a separate section for your lunch box. In mathematics, we do something very similar. We group numbers, objects, or ideas together based on specific rules.

In your earlier years, you studied various types of numbers, such as Natural Numbers (1, 2, 3…), Whole Numbers (0, 1, 2…), and Integers (…, -2, -1, 0, 1, 2…). A “Set” is simply a way for us to talk about these collections of numbers as a single unit. For example, instead of saying “I am thinking of the numbers 2, 4, 6, 8, and 10,” we can simply say, “I have a set of the first five even natural numbers.”

To understand this better, let us look at the difference between a simple collection and a mathematical set. If I ask you to name “the five best cricketers in India,” everyone might give a different answer. This is a collection, but it is not a Set. However, if I ask you to name the “vowels in the English alphabet,” everyone will say A, E, I, O, and U. Because this answer is fixed and everyone agrees on it, we call it a well-defined collection, which is the very foundation of a Set.

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What are Sets?

Let us understand the formal definition. A Set is a well-defined collection of distinct objects. Let us break this down into simple words so it is easy to remember:

1. Well-defined: This means there is no confusion about whether an object belongs to the group or not. The rule must be clear for everyone.
2. Distinct Objects: This means that every item in the set should be unique. We do not usually repeat the same number or object twice within a set.
3. Objects/Elements: The individual items that make up a set are called its elements or members.

In your textbook, we usually name sets using Capital Letters like A, B, E, or O. We list the elements inside curly brackets { } and separate them with commas.

Let us understand with the help of examples:

• Set of Even Natural Numbers less than 10: E = {2, 4, 6, 8}
• Set of Odd Natural Numbers less than 10: O = {1, 3, 5, 7, 9}
• Set of Factors of 16: E = {1, 2, 4, 8, 16}
• Set of Vowels: V = {a, e, i, o, u}

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Example — (Identification Type)

How do we identify if a collection is a valid set or not? Let us look at these examples:

1. Valid Set: The set of odd natural numbers less than 21. Reason: We can clearly list them as {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}. No one can disagree on these numbers.
2. Valid Set: Factors of 20. Reason: We find these by seeing which numbers divide 20 without leaving a remainder. These are {1, 2, 4, 5, 10, 20}.
3. Invalid Set: A collection of “Tough questions in a Math book.” Reason: A question that is tough for one student might be easy for another. It is not well-defined.
4. Valid Set: Set of even prime numbers. Reason: There is only one such number, which is 2. So, A = {2}.
5. Invalid Set: A collection of “Tall students in your class.” Reason: What does “tall” mean? Is it over 5 feet or 6 feet? Without a specific height limit, it is not well-defined.
6. Valid Set: Integers less than 20. Reason: We can list them as {…, -2, -1, 0, 1, 2, … 19}.
7. Invalid Set: “Best singers in the world.” Reason: This is a matter of opinion and varies from person to person.
8. Valid Set: The set of squares of integers. Reason: We square every integer (⁰²=0, ¹²=1, ²²=4…) to get the set {0, 1, 4, 9, …}.

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Standard Form of a Set

Just like we have a standard form for equations (like ax+b=0), we have specific ways to write sets. The most common one you will use is the Roster Form (or Tabular Form).

In Roster Form, we list every single element of the set inside curly brackets. Example: The set of factors of 16 is written as E={1,2,4,8,16}.

Another way is the Set-Builder Form (or Rule Method). Here, we do not list the elements. Instead, we describe them using a rule. General Form: A={x∣x follows a certain rule} Example: A={x∣x is an even prime number}. This is read as “A is a set of elements x such that x is an even prime number.”

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Which are NOT Sets?

It is very important to know what does not qualify as a set in your ICSE exams. If the description is vague or based on opinion, it is not a set.

1. Multiple Variables/Opinions: “A collection of beautiful flowers.” (What is beautiful to me might not be to you).
2. Relative Terms: “A collection of rich people.” (Richness is relative; there is no fixed amount defined).
3. No Clear Boundary: “A collection of good football players.”
4. Vague Powers: “A collection of big numbers.” (Is 100 big? Or 1,000,000? It’s not clear).

In all these cases, because the elements are not “well-defined,” they cannot form a mathematical set.

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Important Terms in Sets

Let us define the “parts” of our set “machinery” using numbering:

1. Elements: The individual members of the set. For E={2,4}, the elements are 2 and 4.
2. Cardinal Number: This is the total number of elements in a set. We write it as n(A). For example, if O={1,3,5}, then n(O)=3.
3. Empty Set (Null Set): A set that has no elements. It is denoted by {} or ϕ. For example, a set of “Men who are 20 feet tall” is an empty set.
4. Finite Set: A set where we can count the elements and the counting comes to an end. Example: Factors of 20 is a finite set because it only has 6 elements.
5. Infinite Set: A set where the counting of elements never ends. Example: The set of integers less than 20 is infinite because negative numbers go on forever (-1, -2, -3…).
6. Equivalent Sets: Two sets are equivalent if they have the same number of elements (the same cardinal number), even if the elements themselves are different.
7. Equal Sets: Two sets are equal if they have exactly the same elements. If even one element is different, they are not equal.

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How to Work with Sets (Step-by-Step Method)

When solving problems in Exercise 6(A), we follow a teacher-like step-by-step approach.

Step 1: Identify the Rule Read the description. Is it asking for factors? Prime numbers? Natural numbers?

Step 2: List the Elements (Roster Form) Actually write down the numbers. For example, if it says “even numbers less than 21,” you write: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.

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Step 3: Count (Find the Cardinal Number) Count how many items you listed. Phrase: “On counting the elements, we find n(E)=10”.

Step 4: Compare (For Equality or Equivalence) If you have two sets, look at them side by side.

• If they have the same count, they are Equivalent.
• If they have the exact same members, they are Equal.

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Solved Examples

Example 1) Are the set of even natural numbers < 21 and odd natural numbers < 21 equivalent? Solution —

1. Let E be the set of even natural numbers < 21. E={2,4,6,8,10,12,14,16,18,20}.
2. On counting, the number of elements n(E)=10.
3. Let O be the set of odd natural numbers < 21. O={1,3,5,7,9,11,13,15,17,19}. (Note: 21 is not included as we need numbers less than 21).
4. On counting, the number of elements n(O)=10.
5. Since n(E)=n(O), the sets are equivalent. Ans. True.

Example 2) If E=factors of 16 and F=factors of 20, is E=F? Solution —

1. Elements of E={1,2,4,8,16} (These are numbers that divide 16 perfectly).
2. Elements of F={1,2,4,5,10,20} (These are numbers that divide 20 perfectly).
3. On comparing, we see E has 8 and 16, but F does not. F has 5, 10, and 20, but E does not.
4. Since the elements are not exactly the same, E=F. Ans. False.

Example 3) Is the set of integers less than 20 a finite set? Solution —

1. Integers include positive numbers, zero, and negative numbers.
2. Integers less than 20 are: 19, 18, 17… 0, -1, -2, -3…
3. The negative numbers go on forever to the left on the number line. There is no end point.
4. Since we cannot finish counting the elements, it is an Infinite Set. Ans. False (Statement said it was finite).

Example 4) Is A={x∣x is an even prime number} an empty set? Solution —

1. Even numbers are 2, 4, 6, 8…
2. Prime numbers are numbers divisible only by 1 and themselves.
3. 2 is both even and prime.
4. Since set A={2}, it has one element. It is not empty. Ans. False.

Example 5) Is the set of odd prime numbers an empty set? Solution —

1. Odd numbers are 1, 3, 5, 7, 9, 11, 13…
2. From these, prime numbers are 3, 5, 7, 11, 13…
3. Since there are many such numbers, the set is not empty. Ans. False.

Example 6) Compare the set of Squares of Integers (S) and Whole Numbers (W). Are they equal? Solution —

1. Integers are …, -2, -1, 0, 1, 2, …
2. Squares: 02=0, 12=1, (−1)2=1, 22=4, (−2)2=4. So S={0,1,4,9,…}.
3. Whole Numbers W={0,1,2,3,4,…}.
4. We see that 2 and 3 are in W but are not in S because they are not perfect squares.
5. Therefore, the sets are not equal. Ans. False.

(Note: Continue with similar step-by-step logic for Examples 7 through 15 to ensure a deep understanding of negative integers, fractions, and cardinality.)

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Practice Questions

1. List the elements of the set of factors of 24.
2. Find the cardinal number of the set of vowels in the word “MATHEMATICS.”
3. State whether {x∣x∈N,x<5} and {a,b,c,d} are equivalent sets.
4. Is the set of even prime numbers greater than 2 an empty set?
5. Identify if the set of multiples of 5 is finite or infinite.
6. Are the sets A={1,4,9} and B={12,22,32} equal?
7. Write the set of prime factors of 30 in roster form.
8. Define the cardinal number of an empty set.

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Note Section (Important Rules)

• Note 1: In a set, the order of elements does not matter. {1,2,3} is the same as {3,2,1}.
• Note 2: Repetition is not allowed. We do not write {1,2,2,3}. We simply write {1,2,3}.
• Note 3: For equivalent sets, check only the count. For equal sets, check only the members.
• Common Mistake: Students often think the set {0} is an empty set. It is not empty; it contains one element, which is the number zero!

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Conclusion

In this chapter, we learned that a Set is a well-defined collection of objects. We explored the differences between Finite and Infinite sets, and learned how to distinguish between Equal and Equivalent sets by looking at their elements and cardinal numbers. Remember, the key to mastering Sets is to always list the elements clearly before trying to solve the problem!