Sets Class 8th ICSE Mathematics

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Introduction

Welcome, students! In our previous classes, we have often grouped objects together based on certain characteristics. For example, you might have a “collection of storybooks,” a “collection of pens,” or a “collection of cricket players.” In mathematics, this idea of grouping things together is given a special name: a Set.

A set is basically a way to organize information so that we can study it more easily. Just as we use variables and constants in Linear Equations to find unknown values, we use sets to understand the relationships between different groups of numbers or objects. Let us understand this basic concept first. When we talk about a “collection,” it must be very specific. If I say “a collection of good books,” it is not a set because a book that is “good” for me might not be “good” for you. However, if I say “a collection of books in our school library,” every student will identify the same books. This specific nature is what makes a collection a Set.

What are Sets?

A Set is a well-defined collection of objects. The word “well-defined” means that there should be no doubt whether a particular object belongs to the collection or not.

In simple words, if everyone agrees on what belongs in the group, it is a set.

• The objects belonging to a set are called its Elements or Members.
• We usually denote sets with Capital Letters (like A, B, M, N).
• We represent elements of a set, especially if they are letters, with Small Letters.
• Elements are always enclosed within curly brackets { } and separated by commas.

Let us understand with the help of examples:

1. The set of vowels in the English alphabet: V={a,e,i,o,u}.
2. The set of letters in the word “REAL”: M={r,e,a,l}.
3. The set of natural numbers less than 5: A={1,2,3,4}.
4. The set of whole numbers less than 4: B={0,1,2,3}.

Examples (Identification Type)

To identify a set, we must check if the collection is well-defined.

Valid Examples (These ARE Sets):

1. Rivers in India: Everyone can list the same rivers (Ganga, Yamuna, etc.).
2. Even numbers between 1 and 10: These are exactly 2, 4, 6, and 8.
3. Letters of the word “LARE”: These are specifically l, a, r, and e.
4. Natural numbers **x such that 3<x≤6:** These are exactly 4, 5, and 6.
5. Whole numbers less than 6: These are 0, 1, 2, 3, 4, and 5.

Invalid Examples (These ARE NOT Sets):

1. Collection of intelligent students in Class 8: “Intelligent” is not well-defined; it is a matter of opinion.
2. Collection of beautiful flowers: Beauty is subjective.
3. Collection of difficult math problems: A problem might be difficult for one student but easy for another.
4. Collection of tall people in your city: “Tall” is not a fixed measurement.
5. Collection of best cricket players: “Best” depends on individual choice.

General Form and Representation

Just like a linear equation has a standard form (ax+b=0), sets have two main ways of being written:

1. Roster Form (or Tabular Form): In this form, we list all the elements explicitly within curly brackets.

• Example: Set of first three natural numbers = {1,2,3}.

1. Set-Builder Form (or Rule Method): In this form, we do not list the elements. Instead, we write a rule or property that all elements must follow.

• Example: A={x:x is a natural number and 3<x≤6}.
• To convert this to Roster Form, we find numbers that fit the rule: x must be greater than 3 (so 4, 5…) and less than or equal to 6. Thus, A={4,5,6}.

Which are NOT Sets? (Common Mistakes)

Students often make mistakes by including collections that are not well-defined. Remember, if the elements can change from person to person, it is not a set.

• Rich people in India: There is no specific limit to what “rich” means.
• Honest students in a class: Honesty is a quality that cannot be measured mathematically.
• Famous actors: “Famous” depends on who you ask.
• Delicious fruits: Taste varies for everyone.

Parts and Important Terms

1. Elements: The individual objects in a set. If x is in set A, we say x belongs to A.
2. Cardinal Number: The total number of unique elements in a set.
3. Empty Set (or Null Set): A set with no elements, denoted by ϕ or {}. For example, the intersection of two sets with no common elements results in an empty set.
4. Subset (⊂): Set A is a subset of B if every element of A is also present in B.
5. Equal Sets (=): Two sets are equal if they have exactly the same elements, even if the order is different.
6. Union (∪): A new set containing all elements from both sets, without repeating any element.
7. Intersection (∩): A new set containing only the common elements present in both sets.
8. Difference ($- $): A−B means the set of elements that are in A but not in B.

How to Solve Set Operations (Step-by-Step)

Let us learn how to perform operations like Union and Intersection.

1. Finding the Union (**A∪B):**

• Step 1: List all elements of Set A.
• Step 2: List all elements of Set B.
• Step 3: Combine them into one set.
• Step 4: On removing duplicates, ensure no element is written twice.

2. Finding the Intersection (**A∩B):**

• Step 1: Look at Set A and Set B.
• Step 2: Identify elements present in both sets.
• Step 3: Write only these common elements in curly brackets.

3. Finding the Difference (**A−B):**

• Step 1: Start with all elements of Set A.
• Step 2: Check if any elements of Set B are inside Set A.
• Step 3: On subtracting/removing those elements of B from A, write the remaining elements of A.

Solved Examples

Example 1) Express set M={letters of the word REAL} in Roster Form. Solution — The word “REAL” has letters R, E, A, and L. When writing elements, we use small letters. So, M={r,e,a,l}. Ans. M={r,e,a,l}.

Example 2) If M={r,e,a,l} and N={l,a,r,e}, is M a subset of N? Solution — To be a subset, every element of M must be in N. r∈N? Yes. e∈N? Yes. a∈N? Yes. l∈N? Yes. Since all elements of M are in N, M⊂N. Ans. Yes.

Example 3) Given A={x:x∈N,3<x≤6}, find the Roster Form. Solution — Natural numbers (N) start from 1. We need numbers x such that x>3 and x≤6. The numbers are 4, 5, and 6. Ans. A={4,5,6}.

Example 4) Find the union of A={4,5,6} and B={0,1,2,3}. Solution — **A∪B includes all elements from both. A∪B={4,5,6,0,1,2,3}. Ans.* A∪B*={0,1,2,3,4,5,6}.

Example 5) Find the intersection of A={4,5,6} and B={0,1,2,3}. Solution — Identify common elements. Are there any numbers present in both sets? No. Since there are no common elements, the set is empty. Ans. A∩B=ϕ.

Example 6) If P={4,5,6,7,8} and Q={1,2,3,4,5}, find P∩Q. Solution — Common elements are 4 and 5. Ans. P∩Q={4,5}.

Example 7) For the same sets P and Q above, find P∪Q. Solution — Combine elements: {4,5,6,7,8} and {1,2,3,4,5}. On transposing them into one set and removing repeats: {1,2,3,4,5,6,7,8}. Ans. P∪Q={1,2,3,4,5,6,7,8}.

Example 8) Calculate A−B if A={4,5,6} and B={0,1,2,3}. Solution — We must remove elements of B from A. Since no element of B is in A, nothing is removed. A−B={4,5,6}. Ans. A−B={4,5,6}.

Example 9) If C={1,2,3,4,5} and B={4,5,6,7}, find B∩C. Solution — The common elements between B and C are 4 and 5. Ans. B∩C={4,5}.

Example 10) Verify the Associative Law: (A∪B)∪C=A∪(B∪C) for given sets. Solution —

• LHS: First find (A∪B), then its union with C.
• RHS: First find (B∪C), then the union of A with that result.
• On comparing both sets, if elements are exactly the same, the law is verified. Ans. Yes, LHS = RHS.

Practice Questions

1. Identify if the following is a set: “A collection of the five best cricketers in the world.”
2. Write the set of letters in the word “SCHOOL” in Roster Form.
3. If X={1,2,3} and Y={3,2,1}, are these sets equal?
4. Given A={x:x∈W,x<5}, write A in Roster Form.
5. Find A∪B if A={a,b,c} and B={c,d,e}.
6. Find A∩B for the sets in question 5.
7. If P={10,20,30} and Q={20,40}, find P−Q.
8. Is the set of whole numbers between 2 and 3 a Null Set?

Note Section (Important Rules)

• Note 1: In Roster Form, the order of elements does not matter. {1,2,3} is the same as {3,2,1}.
• Note 2: Do not repeat elements in a set. If a letter appears twice in a word, write it only once in the set.
• Note 3: Always use curly brackets { } for sets. Using square or round brackets is mathematically incorrect.
• Note 4: The intersection of a set with an empty set is always an empty set (A∩ϕ=ϕ).

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Conclusion

In this chapter, we learned that a Set is a well-defined collection of objects. We explored how to represent sets in Roster and Set-Builder forms and how to perform basic operations like Union, Intersection, and Difference. Understanding these basics is essential because sets form the foundation for many advanced topics in mathematics. Keep practicing the step-by-step methods for operations, and you will find sets to be one of the most logical and interesting chapters in your syllabus!