Sets: Roster Form, Set-Builder Form, and Types of Sets — Class 8th

Introduction

Dear students, welcome to the fascinating world of Sets. In your previous classes, you have learned about various collections of numbers, such as Natural Numbers (1,2,3…), Whole Numbers (0,1,2,3…), and Integers (…−2,−1,0,1,2…). In mathematics, we often need to group these numbers or other objects together based on a common rule. This grouping is what we call a Set.

Let us understand this with the help of examples. If I ask you to make a list of all the vowels in the English alphabet, you will quickly write: a,e,i,o,u. This is a clearly defined collection because everyone will give the same answer. However, if I ask you to list the “three best cricketers in the world,” different people might give different names. Because the second collection is not “well-defined,” it is not a set in the mathematical sense.

In this chapter, we will learn how to write these sets in different ways and how to identify if a set has a limit or goes on forever. Specifically, we will dive deep into Exercise 6(A) of your Selina Publication textbook, focusing on Question 6 and Question 7, which deal with converting descriptions into lists and identifying types of sets.

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What is a Set and How is it Represented?

A Set is a well-defined collection of objects. These objects are called the elements or members of the set. To represent a set, we generally use capital letters like A,B,P, or M.

There are two main ways to write a set:

1. Roster (or Tabular) Form: In this method, we list all the elements of the set one by one, separated by commas, and enclose them within curly brackets {}. For example: A={2,4,6,8}.
2. Set-Builder Form: Instead of listing elements, we describe the common property that all elements share. We use a variable (usually x) to represent the elements. For example: A={x:x is an even natural number less than 10}.

In the following sections, we will learn how to convert from Set-Builder form to Roster form by solving mathematical equations and inequalities.

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Examples (Identification Type)

Let us look at some examples to identify valid sets and their forms:

1. {2,3,5,7}: This is a valid set in Roster Form containing prime numbers less than 10.
2. {x:x is a factor of 10}: This is a valid set in Set-Builder Form.
3. The collection of “Difficult Chapters” in Math: This is NOT a set because “difficult” is not well-defined; what is difficult for one student might be easy for another.
4. {a,e,i,o,u}: A valid set of vowels.
5. The collection of “Tall Boys” in your class: This is NOT a set unless we specify a height (e.g., boys taller than 160 cm).
6. {0,1,2,3,4,5}: A valid set of the first six whole numbers.
7. {x:x∈N,x<5}: A valid set-builder form for natural numbers less than 5.
8. The collection of “Honest People” in a city: This is NOT a set as honesty cannot be measured mathematically.

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General Form / Standard Form

In Set-Builder Form, we follow a standard structure:

{x:Condition or Property of x}

• x: This is the variable that represents every element in the set.
• : (Colon): This symbol is read as “such that.” Sometimes a vertical bar ∣ is used instead.
• Condition: This is the mathematical rule or description that x must follow.
• Domain: We often specify which number system x belongs to, such as x∈N (Natural Numbers) or x∈Z (Integers).

Example 1: {x:x∈W,x❤} means “the set of all x such that x is a whole number less than 3.” Example 2: {x:x2=9} means “the set of all numbers whose square is 9.”

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Which are NOT Valid Sets or Correct Forms?

It is important to know what mistakes to avoid:

1. Multiple Variables without definition: {x,y:x+y=5} is confusing if we don’t know what x and y are.
2. Vague Adjectives: As mentioned, “Popular singers,” “Good books,” or “Funny jokes” are not sets.
3. Missing Braces: Writing 1,2,3 instead of {1,2,3} is not a formal set representation.
4. No Commas: {123} is incorrect; you must use commas to separate elements.

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Parts and Important Terms

Let us break down the terms we will use while solving our exercise:

1. Variable: A symbol like x,y, or n used to represent an unknown value in a set-builder rule.
2. Constant: A fixed value, like the numbers 2,5, or 10 in an equation.
3. LHS and RHS: “Left Hand Side” and “Right Hand Side” of an equation or inequality.
4. Natural Numbers (N): Counting numbers starting from 1,2,3….
5. Whole Numbers (W): Numbers starting from 0,1,2,3….
6. Integers (Z or I): Positive numbers, negative numbers, and zero.
7. Finite Set: A set where we can count the number of elements.
8. Infinite Set: A set where the elements never end, and we cannot count them.

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How to Solve: Converting to Roster Form

To convert a set from Set-Builder form to Roster form, follow these steps:

1. Identify the Domain: Look at whether x is a Natural number (N), Whole number (W), or Integer (Z).
2. Write the Equation/Inequality: Focus on the mathematical rule provided.
3. Perform Transposition: Move numbers from one side to another to isolate the variable.

• On transposing a positive number, it becomes negative on the other side.
• On transposing a negative number, it becomes positive.

1. Balance the Operation:

• On dividing both sides by a number to find the value of x.
• On adding/subtracting the same value from both sides to keep the equation balanced.

1. Solve and List: Find the values of x that satisfy the condition and write them inside {}.

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Solved Examples (From Selina Exercise 6A)

Let us solve these step-by-step, just like we do in the classroom.

Example 1) Set-Builder to Roster (Quadratic Equation)

Question: Write in Roster Form: {x:x2−2x−3=0}.

Solution:

1. We have the equation x2−2x−3=0.
2. To solve this, we need to find two integers a and b such that their product is −3 and their sum is −2.
3. Those numbers are −3 and 1, because (−3)×1=−3 and (−3)+1=−2.
4. On splitting the middle term: x2−3x+x−3=0.
5. On taking common factors: x(x−3)+1(x−3)=0.
6. This gives us (x−3)(x+1)=0.
7. So, either x−3=0 (which means x=3) or x+1=0 (which means x=−1).
8. The elements are −1 and 3.

Ans. {−1,3}.

Example 2) Using a Formula with a Range

Question: {x:x=2y+5,y∈N and 2≤y<6}.

Solution:

1. First, identify the values of y. Since y is a natural number between 2 (inclusive) and 6 (exclusive), y can be 2,3,4,5.
2. Now, substitute each y value into the formula x=2y+5:

• When y=2,x=2(2)+5=4+5=9.
• When y=3,x=2(3)+5=6+5=11.
• When y=4,x=2(4)+5=8+5=13.
• When y=5,x=2(5)+5=10+5=15.

1. The set contains these calculated x values.

Ans. {9,11,13,15}.

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Example 3) Factors of a Number

Question: {x:x is a factor of 24}.

Solution:

1. We need to find all numbers that divide 24 exactly without a remainder.
2. Factors of 24 are: 1,2,3,4,6,8,12,24.

Ans. {1,2,3,4,6,8,12,24}.

Example 4) Squares and Inequalities

Question: {x:x∈Z,x2≤4}.

Solution:

1. We need integers whose square is 4 or less than 4.
2. Let us check integers:

• 02=0 (Less than 4 — Yes).
• 12=1 (Less than 4 — Yes).
• 22=4 (Equal to 4 — Yes).
• (−1)2=1 (Less than 4 — Yes).
• (−2)2=4 (Equal to 4 — Yes).
• 32=9 (Greater than 4 — No).

1. The values are 0,1,2,−1,−2.

Ans. {0,1,2,−1,−2}.

Example 5) Solving a Linear Inequality

Question: {x:3x−2≤10,x∈N}.

Solution:

1. The inequality is 3x−2≤10.
2. On adding 2 to both sides: 3x−2+2≤10+2⇒3x≤12.
3. On dividing both sides by 3: x≤12/3⇒x≤4.
4. Since x must be a natural number (N), the values are 1,2,3,4.

Ans. {1,2,3,4}.

Example 6) Multiplying by Negative Numbers

Question: {x:4−2x>−6,x∈Z}.

Solution:

1. The inequality is 4−2x>−6.
2. On subtracting 4 from both sides: −2x>−6−4⇒−2x>−10.
3. Note: When we multiply or divide by a negative number, the inequality sign reverses.
4. On dividing by -2: x<(−10)/(−2)⇒x<5.
5. Since x is an integer (Z) less than 5, the set includes 4,3,2,1,0,−1,−2… and goes on forever.

Ans. {…,−2,−1,0,1,2,3,4}.

Example 7) Identifying Infinite Sets

Question: Is {x:x∈Z,x<10} finite or infinite?.

Solution:

1. We need integers less than 10.
2. These are 9,8,7,… and it goes on to negative infinity (…−100,−101…).
3. Since we cannot count all the elements, it is an infinite set.

Ans. Infinite Set..

Example 8) Finite Whole Numbers

Question: Is {x:x∈W,5x−3≤20} finite or infinite?.

Solution:

1. Solve 5x−3≤20.
2. On transposing -3: 5x≤20+3⇒5x≤23.
3. On dividing by 5: x≤23/5⇒x≤4.6.
4. Since x is a whole number, x can be 0,1,2,3,4.
5. There are exactly 5 elements. We can count them.

Ans. Finite Set..

Example 9) Infinite Natural Numbers

Question: Is P={y:y=3x−2,x∈N,x>5} finite or infinite?.

Solution:

1. x are natural numbers greater than 5 (6,7,8,9… forever).
2. Since there are infinite values for x, when we put them into the formula y=3x−2, we will get infinite values for y.

Ans. Infinite Set..

Example 10) Fraction-Based Sets

Question: M={x:x=3/n,n∈W,6<n≤15}. Is it finite?.

Solution:

1. Identify values for n: 7,8,9,10,11,12,13,14,15.
2. Calculate x for each n: x=3/7,3/8,3/9(1/3),3/10,….
3. There are 9 distinct values. We can count them.

Ans. Finite Set..

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Practice Questions

Now it is your turn to practice! Solve the following in your notebook:

1. Write in Roster form: {x:x∈N,2x+3=11}.
2. Write in Roster form: {x:x is a letter in the word “MATHEMATICS”}.
3. Write in Roster form: {x:x∈Z,−2<x≤3}.
4. Identify as Finite or Infinite: {x:x is a multiple of 5}.
5. Identify as Finite or Infinite: {x:x∈W,x2<50}.
6. Write in Roster form: {x:x=n2,n∈N,n≤4}.
7. Solve and write in Roster: {x:5−x>2,x∈N}.
8. Identify as Finite or Infinite: The set of all stars in the sky.

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Note Section (Important Rules)

• Rule 1: In Roster form, the order of elements does not matter. {1,2} is the same as {2,1}.
• Rule 2: Elements are never repeated. Even if a number appears twice in a calculation, write it once in the set.
• Rule 3: Crucial Inequality Rule! Always remember to reverse the inequality sign if you multiply or divide both sides by a negative number.
• Rule 4: Always check your domain (N,W, or Z) before listing elements.

Conclusion

In this lesson, we learned how to translate the “language” of mathematics (Set-Builder form) into a simple list (Roster form) by solving equations and inequalities. We also learned how to distinguish between sets that have an end (Finite) and sets that go on forever (Infinite). Mastering these basics will help you solve more complex problems in Algebra and Probability. Keep practicing, and remember to always balance your equations!