SETS Class 8th ICSE Maths / Selina Chapter 6

Introduction

In our previous classes, we have often grouped objects together based on certain similarities. For example, we might talk about a “collection of books,” a “bunch of keys,” or a “team of players.” In mathematics, this idea of grouping things together is given a very specific name: a Set. A set is more than just a random collection; it is a way to organize information so that we can study the relationships between different groups.

Before we dive deep into the world of sets, let us remember some basic concepts. We already know about different types of numbers, like Natural Numbers (1, 2, 3…) and Integers (…, -2, -1, 0, 1, 2…). We also know about geometric shapes like Quadrilaterals, which are closed figures made of four sides. Just as we use variables and constants to form linear equations, we use “elements” to form sets. Understanding how these elements relate to each other is the foundation of this chapter.

What are Sets?

A Set is a well-defined collection of objects. The word “well-defined” is very important here. It means that there should be no doubt or confusion about whether a particular object belongs to the collection or not. Every object that belongs to a set is called an element or a member of that set.

Let us understand this with a simple explanation. If I say “the set of all vowels in the English alphabet,” everyone will agree that the members are {a, e, i, o, u}. Because there is no confusion, it is a set. However, if I say “the collection of three best cricketers in the world,” different people will have different opinions. Since it is not clearly defined for everyone, it is not a mathematical set.

Conditions for a Set:

1. Well-defined: The rule for belonging must be clear.
2. Distinct Elements: Usually, elements in a set are not repeated.
3. Order does not matter: The order in which we list elements does not change the set.

Examples of Sets:

1. Set of even numbers less than 10: {2, 4, 6, 8}.
2. Set of primary colors: {Red, Yellow, Blue}.
3. Set of all Quadrilaterals (closed figures with four sides).
4. Set of all Squares.
5. Set of all Integers between -2 and 2: {-1, 0, 1}.

Examples (Identification Type)

Let us look at some examples to identify what is a set and what is not.

1. The collection of all months in a year. (Valid: Everyone knows there are 12 specific months).
2. The collection of all rich people in Delhi. (Invalid: “Rich” is not well-defined; it varies from person to person).
3. Set A = {Rectangles, Squares, Rhombuses}. (Valid: These are specific types of quadrilaterals).
4. The collection of intelligent students in your class. (Invalid: “Intelligent” is a subjective term).
5. The collection of whole numbers less than 5. (Valid: The elements are 0, 1, 2, 3, 4).
6. The collection of delicious fruits. (Invalid: Taste is a matter of opinion).
7. Set B = {x | x is a natural number and 10 ≤ x ≤ 15}. (Valid: It clearly defines numbers 10, 11, 12, 13, 14, 15).
8. The collection of tall boys in the school. (Invalid: “Tall” is not precisely defined).
9. The collection of planets in our solar system. (Valid: There are 8 specific planets).
10. Set of all integers greater than -6 and less than or equal to 6. (Valid: These are integers from -5 to 6).

General Form / Representation of Sets

Just as a linear equation has a standard form like ax+b=0, a set can be represented in two main ways.

1. Roster or Tabular Form: In this form, we list all the elements of the set inside curly brackets { } and separate them by commas.

• Example: A set of first five natural numbers is A = {1, 2, 3, 4, 5}.

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. It is written as: A={x∣x has property P} Here, ‘x’ is a variable representing the elements, and the vertical bar ‘|’ or colon ‘:’ means “such that.”

• Example 1: U={x∣x∈Natural Numbers and 10≤x≤35}. This represents natural numbers from 10 to 35.
• Example 2: N={n∣n is a non-negative number in the Universal set}. This describes numbers that are not negative, like 0, 1, 2, 3….

Which are NOT Sets?

It is very important to recognize what cannot be called a set in mathematics. If a collection uses “adjectives” that depend on personal opinion, it is NOT a set.

1. Collection of beautiful flowers: “Beautiful” is subjective; different people like different flowers.
2. Collection of difficult chapters in Maths: What is difficult for one student might be easy for another.
3. Collection of good football players: “Good” is not a mathematical measurement.
4. Collection of interesting books: “Interesting” is not well-defined.
5. Collection of big numbers: There is no definition for where a number starts being “big.”
6. Collection of talented singers: “Talent” is an opinion, not a fact.

Parts and Important Terms

To solve problems in Chapter 6, you must understand these five terms:

1. Universal Set (U): This is the “mother” set that contains all the elements under consideration for a particular problem.
2. Element: An individual object belonging to a set. For example, in Set A = {1, 2}, 1 is an element.
3. Subset (⊆): Set B is a subset of Set A if every element of B is also in A. For example, all squares are rectangles, so the set of squares is a subset of the set of rectangles.
4. Proper Subset (⊂): Set D is a proper subset of Set A if D is a subset of A, but A has at least one extra element that is not in D. For example, the set of Rhombuses is a proper subset of Quadrilaterals because every rhombus is a quadrilateral, but there are other quadrilaterals (like rectangles) that are not rhombuses.
5. Complement of a Set (A’): The complement of set A consists of all elements in the Universal set (U) that are not present in set A.

How to Solve Set Problems

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When solving problems involving subsets or complements, we follow a very disciplined step-by-step method, much like solving equations.

Step 1: Convert to Roster Form If the sets are given in Set-Builder form (using rules), our first step is to list the elements clearly. This makes it easier to compare them.

Step 2: Compare Elements To check if a set is a subset, we look at each element one by one. If even one element is missing from the bigger set, it is not a subset.

Step 3: Finding the Complement To find the complement (A’), follow these teaching phrases:

• “Identify the Universal set (U).”
• “Identify Set A.”
• “On removing all elements of A from U, the remaining elements form A’.”

Solved Examples

Let us understand the concepts with the following detailed examples:

Example 1) If Set B is the set of all Rectangles and Set C is the set of all Squares, is B a subset of C? Solution:

• A square must have all four sides equal and all angles 90°.
• A rectangle only needs opposite sides equal and all angles 90°.
• While every square is a rectangle, not all rectangles are squares because a rectangle might have unequal adjacent sides.
• Since all elements of Set B are not in Set C, Set B is not a subset of Set C. Ans. False.

Example 2) If Set C = {Squares} and Set B = {Rectangles}, is C a subset of B? Solution:

• A rectangle must have angles of 90° and opposite sides equal.
• A square has all angles 90° and all sides equal. This means its opposite sides are definitely equal.
• Therefore, every square fulfills the criteria to be a rectangle.
• Since every element of C is in B, C ⊆ B. Ans. True.

Example 3) Let Universal Set U = {10, 11, 12, …, 35}. If Set A = {x | x is a natural number ≤ 16}, find A’. Solution:

• First, write A in roster form: A = {1, 2, 3, …, 16}.
• Universal set starts from 10: U = {10, 11, 12, 13, 14, 15, 16, 17, …, 35}.
• The elements 10 to 16 are present in both U and A.
• On subtracting elements of A from U, we are left with numbers starting from 17 up to 35.
• A’ = {17, 18, 19, …, 35}. Ans. A’ = {x | x ∈ N, 17 ≤ x ≤ 35}.

Example 4) U = {Integers from -5 to 6}. Set N = {Non-negative numbers}. Find N’. Solution:

• U = {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6}.
• Non-negative numbers are those that are not negative, which includes zero: N = {0, 1, 2, 3, 4, 5, 6}.
• On removing elements of N from U, we are left with the negative integers.
• N’ = {-5, -4, -3, -2, -1}. Ans. N’ = {-5, -4, -3, -2, -1}.

Example 5) U = {Integers from -5 to 6}. Set P = {Non-positive numbers}. Find P’. Solution:

• U = {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6}.
• Non-positive means not positive. This includes zero and negative numbers: P = {0, -1, -2, -3, -4, -5}.
• On removing elements of P from U, we are left with the positive numbers 1 to 6.
• P’ = {1, 2, 3, 4, 5, 6}. Ans. P’ = {1, 2, 3, 4, 5, 6}.

(Note: Additional examples 6–10 follow the same logic of identifying elements and comparing sets.)

Practice Questions

1. Identify if the collection of “best storybooks” is a set. Why?
2. If U = {Natural numbers less than 20} and A = {Multiples of 5}, find A’.
3. Write the set {2, 4, 6, 8, 10} in Set-Builder form.
4. If Set A = {Rhombuses} and Set B = {Quadrilaterals}, is A a proper subset of B?
5. If x is an integer and −3<x≤2, list the elements of the set in Roster form.
6. Find the complement of the set of even numbers if the Universal set is the set of first 10 natural numbers.
7. True or False: Every rectangle is a square. Explain why.
8. Define a Universal set for the sets A = {Cat, Dog} and B = {Lion, Tiger}.

Note Section (Important Rules)

• Note 1: Always check the Universal Set before listing elements. Elements of any subset must come from the Universal set.
• Note 2: The number 0 is neither positive nor negative. Therefore, it is included in both “non-negative” and “non-positive” sets.
• Note 3: Use curly brackets { } only. Using square or round brackets for sets is a common mistake.
• Note 4: When finding the complement, ensure the elements you list are actually present in the Universal set.

Conclusion

Sets are a fundamental part of mathematics that help us categorize and compare information. By mastering the conversion between Roster and Set-Builder forms and understanding the relationship between sets (like subsets and complements), you build a strong foundation for higher mathematics. Remember, a set is only a set if it is well-defined! Keep practicing the step-by-step method of listing elements to avoid confusion.