SETS Class 8th ICSE Maths (Selina Chapter 6)

Introduction

Welcome, students! In your previous classes, you have often grouped things together. For example, you might have a collection of your favorite storybooks, a collection of different colored pens, or a group of your best friends. In mathematics, we have a very special and precise way of talking about these groups or collections. We call them Sets.

Before we dive deep, let us understand some basic concepts. A collection is simply a group of objects. However, in math, not every group is a “Set.” For a group to be called a Set, it must be well-defined. This means that if I ask you and your friend to list the items in that group, both of you should come up with the exact same list. For instance, the “collection of the first five natural numbers” will always be 1, 2, 3, 4, and 5, no matter who writes it. This is a well-defined collection. In this chapter, we will learn how to identify, write, and solve problems related to these sets using the Selina ICSE Mathematics approach.

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What are Sets?

A Set is a well-defined collection of distinct objects. Let us break this definition into small parts to understand it clearly:

1. Well-defined: This is the most important condition. It means there is no confusion about whether an object belongs to the group or not. It should not depend on anyone’s opinion.
2. Distinct objects: This means the objects in a set should be different from each other. We usually do not repeat the same number or letter inside a set.
3. Elements: The objects that make up a set are called its elements or members.

How do we write a set?

• We always use a Capital Letter (like A,B,S, or U) to name a set.
• We use Curly Brackets {} to enclose the elements.
• We separate the elements using commas.

Let us understand with the help of examples:

• A={1,2,3,4,5} (The set of first five natural numbers)
• V={a,e,i,o,u} (The set of vowels in the English alphabet)
• P={2,3,5,7} (The set of prime numbers less than 10)

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Examples (Identification Type)

How do we identify if a collection is a valid Set or an invalid one? Let’s look at 10 examples:

1. The collection of all vowels in the word “MATHEMATICS”: Valid. Everyone will agree the vowels are A, E, and I.
2. The collection of the first three months of a year: Valid. These are January, February, and March.
3. The collection of “Difficult” chapters in Math: Invalid. A chapter that is difficult for you might be easy for your friend. It is based on opinion.
4. The collection of all factors of 12: Valid. The factors are 1, 2, 3, 4, 6, and 12.
5. The collection of “Tall” boys in your class: Invalid. “Tall” is not defined. One person might think 5 feet is tall, while another might think 6 feet is tall.
6. The collection of even numbers between 1 and 10: Valid. These are 2, 4, 6, and 8.
7. The collection of “Good” cricketers in India: Invalid. “Good” is a matter of choice and opinion.
8. The collection of states in India: Valid. This is a fixed, well-defined list.
9. The collection of “Famous” scientists: Invalid. Fame is subjective; it changes from person to person.
10. The collection of natural numbers less than 20: Valid. As discussed in our sources, these are numbers from 1 to 19.

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General Form / Representation of Sets

In mathematics, we represent sets in two main ways. Think of this like a “Standard Form” for sets.

1. Roster or Tabular Form: In this form, we list all the elements one by one inside curly brackets.

• Example: A={2,4,6,8,10}

2. Set-Builder Form: In this form, we do not list the elements. Instead, we write a rule or a property that all elements must follow. The general form looks like this: A={x∣x satisfies a property} The symbol “|” or “:” is read as “such that.”

• Example: U={x∣x∈N,x<20}.
• Here, x represents the elements.
• The rule says x is a Natural Number (N) and x is less than 20.
• So, in Roster form, U={1,2,3,4,…,19}.

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Which are NOT Sets?

It is very important to know what does not count as a set. If the description is vague or depends on personal taste, it is not a set.

1. A collection of “Interesting” books: Different people find different things interesting.
2. A collection of “Talented” singers: Talent is a subjective quality.
3. A collection of “Big” numbers: What is big? 1,000? 1,000,000? It is not defined.
4. A collection of “Old” people in a city: “Old” could mean over 60, over 70, or over 80.
5. A collection of “Honest” students: Honesty cannot be measured by a mathematical rule.
6. A collection of “Tasty” fruits: Taste is a personal preference.

Reasoning: All these examples fail the “well-defined” test. Because the criteria (interesting, big, tasty) are not fixed, these collections are not sets.

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Parts and Important Terms

To solve problems like a pro, you must know these terms:

1. Universal Set (U): This is the “Father Set.” It contains all the possible elements we are considering for a particular problem.
2. Empty Set / Null Set (∅): A set that has no elements at all. It is represented by the symbol ∅ or empty brackets {}.
3. Subset (⊆): Set A is a subset of B if every element of A is also found in B.
4. Proper Subset (⊂): Set A is a proper subset of B if A is a subset of B, but A is not equal to B. This means B must have at least one element that is not in A.
5. Cardinal Number n(A): This simply means the number of elements in set A.

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How to Solve Set Problems

Let us learn the step-by-step method to convert a Set-Builder form into a Roster form, which is a common task in your exams.

Step 1: Identify the Rule. Look at the formula given, such as x=3p or y=2n+3.

Step 2: Check the Constraints. Look at what the variables represent (like “p is a natural number”) and check the Universal Set (U).

Step 3: Perform Substitution. Plug in values (1,2,3…) into the formula to find the potential elements.

Step 4: Filter by Universal Set. On comparing with the Universal Set, we only keep the values that actually exist in U. Values outside U are rejected.

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Solved Examples

Example 1) Let U={x∣x∈N,x<20}. Write U in Roster form. Solution: Natural numbers start from 1. The condition is x<20. So, we list numbers from 1 up to 19. U={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19} Ans.

Example 2) Given U from Ex 1, find Set A={x∣x=3p,p∈N}. Solution: We use the formula x=3p and substitute natural numbers for p:

• If p=1,x=3(1)=3
• If p=2,x=3(2)=6
• If p=3,x=3(3)=9
• If p=4,x=3(4)=12
• If p=5,x=3(5)=15
• If p=6,x=3(6)=18
• If p=7,x=3(7)=21 On comparing with U, we see that 21 is not in the Universal set. So, A={3,6,9,12,15,18}. Ans.

Example 3) Find Set B={y∣y=2n+3,n∈N} where elements must be in U. Solution: Using the formula y=2n+3:

• If n=1,y=2(1)+3=5
• If n=2,y=2(2)+3=7
• If n=3,y=2(3)+3=9
• …and so on, until y=19 (when n=8).
• If n=9,y=2(9)+3=21 (Rejected as it’s not in U). B={5,7,9,11,13,15,17,19}. Ans.

Example 4) Set C={x∣x is divisible by 4, x∈U}. Solution: We look for multiples of 4 within the range of U (1 to 19). The multiples are 4, 8, 12, and 16. C={4,8,12,16}. Ans.

Example 5) Solve for Set D={x∣x2−9x−10=0}. Solution: This is a quadratic equation. We must factorize it. We need two numbers that multiply to -10 and add to -9. Those are -10 and 1. x2−10x+x−10=0 x(x−10)+1(x−10)=0 (x−10)(x+1)=0. On transposing, we get x=10 or x=−1. So, D={−1,10}. Ans.

Example 6) List all subsets of Set D={−1,10}. Solution:

1. The Empty Set: ∅
2. Single element sets: {−1},{10}
3. The set itself: {−1,10} Ans..

Example 7) List all proper subsets of Set D={−1,10}. Solution: A proper subset must have fewer elements than the original set. The proper subsets are: ∅,{−1},{10}. (Note: {−1,10} is a subset but not a proper subset because it has the same number of elements). Ans.

Example 8) Is the set of Equilateral Triangles © a subset of Isosceles Triangles (B)? Solution: An isosceles triangle has at least 2 equal sides. An equilateral triangle has all 3 sides equal. Since all 3 sides are equal, it automatically means at least 2 sides are equal. Therefore, all equilateral triangles are isosceles. So, C⊆B is True. Ans.

Example 9) Is the set of Isosceles Triangles (B) a proper subset of all Triangles (A)? Solution:

1. Every isosceles triangle is a triangle (B⊆A).
2. Set A contains other triangles (like scalene) that are not in B. Thus, B has fewer elements than A. So, B⊂A is True. Ans.

Example 10) If C⊂B and B⊂A, then is C⊂A? Solution: Yes. If all equilateral triangles are isosceles, and all isosceles triangles are triangles, then all equilateral triangles are definitely triangles. This is called a transitive relationship. Ans. True.

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Practice Questions

1. Write the set of all factors of 24 in Roster form.
2. Represent A={x∣x is a letter in the word ’SCHOOL’} in Roster form.
3. Given U={1,2,3,4,5,6,7,8,9,10}, find B={x∣x is a prime number}.
4. If S={a,b}, list all its subsets.
5. How many proper subsets does the set {5,6,7} have?
6. True or False: The set of “Best Indian Food” is a well-defined set.
7. Write {2,4,6,8,10} in Set-Builder form.
8. Find the cardinal number of the set V={a,e,i,o,u}.

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Note Section (Important Rules)

• Note 1: Always check your Universal Set (U). You cannot include any element in your answer that is not present in the Universal Set.
• Note 2: Every set is a subset of itself, but it is never a proper subset of itself.
• Note 3: The Empty Set (∅) is a subset and a proper subset of every non-empty set.
• Note 4: The order of elements does not matter. {1,2,3} is the same as {3,2,1}.

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Conclusion

In this chapter, we learned that a Set is a well-defined collection of distinct objects. We explored how to write them in Roster and Set-Builder forms and how to identify subsets and proper subsets. Remember, the key to mastering Sets is to always look for the rule that defines the collection. If the rule is clear and doesn’t change based on opinion, you’ve got yourself a Set! Happy practicing!