Sets, Finite and Infinite Sets, and Equal Sets: Class 8th ICSE Maths

Parul Rajpal

10 min read

·

Apr 7, 2026

--

Listen

Share

Introduction

Hello students! Welcome to our detailed study of Sets. In your previous classes, you have already learned about various types of numbers like Natural Numbers, Whole Numbers, and Integers. You might also remember how we group things together in our daily lives — like a “bunch of keys” or a “team of players.” In Mathematics, this grouping has a special name and a very specific set of rules. This chapter is the foundation for many higher-level math concepts, so it is very important to understand it step-by-step.

Before we dive deep, let us refresh our memory on basic number systems. Natural Numbers (N) start from 1, 2, 3, and go on forever. Whole Numbers (W) are just like natural numbers but they also include 0. Integers (I or Z) include all positive numbers, negative numbers, and zero. We call these “elements” when we put them inside a set. A Set is simply a well-defined collection of these elements. Just like how we use expressions and equations to represent relationships between numbers, we use Sets to organize and compare groups of numbers.

In this chapter, we are going to focus on three very important questions: How do we know if two sets are exactly the same (Equal Sets)? When does a set have an end (Finite Sets)? And when does a set go on forever without stopping (Infinite Sets)? We will use examples from your Selina ICSE Class 8th textbook, specifically focusing on the logic used to solve complex problems involving equations and inequalities within sets.

— — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —

What are Finite, Infinite, and Equal Sets?

Let us understand these terms in very simple words.

1. Finite Sets: A set is called Finite if we can count its members or elements. It has a definite starting point and a definite ending point. For example, the set of whole numbers less than 12 is finite because we can list them all and stop at 11.

2. Infinite Sets: A set is called Infinite if the number of elements is endless. We cannot finish counting them even if we tried for a hundred years! For example, the set of all multiples of 8 is infinite because we can keep multiplying 8 by larger and larger natural numbers (8,16,24,32…) and the list will never end.

3. Equal Sets: Two sets, let’s call them Set A and Set B, are said to be Equal only if they meet two strict conditions:

• They must have the exact same elements.
• They must have the same number of elements (this is called the cardinal number).

Conditions for Equality: Think of it like two pencil boxes. If Box A has a red pen and a blue pen, and Box B also has exactly one red pen and one blue pen, they are equal. But if Box B has a red pen, a blue pen, and a pencil, they are NOT equal because the members are different.

Examples for Understanding:

• Example 1: Set of vowels in “HELLO” -> {E, O}. This is Finite.
• Example 2: Set of stars in the sky. This is Infinite.
• Example 3: Set A = {1, 2} and Set B = {x | x is a natural number less than 3}. These are Equal because B also contains {1, 2}.

— — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —

Examples (Identification Type)

Let us understand how to identify these sets with the help of examples.

1. The set of multiples of 8: Infinite. Since natural numbers (1,2,3…) are endless, the products (8×1,8×2…) are also endless.
2. The set of integers less than 10: Infinite. While there is an upper limit (10), there is no lower limit for negative integers (−1,−2,−3…), so the count never stops.
3. The set of whole numbers less than 12: Finite. We can count them: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}. There are exactly 12 elements.
4. Set A = {2, 4, 6, 8} and Set B = {2, 4, 6, 8}: Equal. Every element is identical.
5. Set of days in a week: Finite. There are exactly 7 days.
6. Set of points on a straight line: Infinite. You can always find a smaller point between two points.
7. Set M = {0, 1, 2, 3, 4} and Set N = {1, 3, 5, 7}: NOT Equal. Set M has 5 elements, but Set N only has 4. Also, the numbers themselves are different.
8. The set of grains of sand on a beach: Finite. (Even though it is a very large number, it is technically countable and has an end, unlike the set of numbers).
9. The set of even prime numbers: Finite. There is only one such number: {2}.
10. The set of months starting with ‘Z’: Finite. This is actually an empty set {}, and all empty sets are finite because the count is zero.

— — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —

General Form / Standard Form

In Mathematics, we often represent sets in a “Set-Builder Form,” which looks like a formula. The standard way to write this is:

{x∣x satisfies a certain condition}

In this form:

• x is the variable representing any element in the set.
• ∣ or : means “such that.”
• The condition tells us which numbers are allowed to enter the set.

Examples of Standard Form:

1. A={x∣x∈N,x<5}: This means x is a natural number and it must be smaller than 5. On solving, we get {1, 2, 3, 4}.
2. B={2n∣n∈N,n<5}: Here, we first find n (1, 2, 3, 4) and then multiply each by 2 to get the set elements {2, 4, 6, 8}.
3. C={y∣y2−3y+2=0}: Here, the elements are the solutions to a quadratic equation.

— — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —

Which are NOT Equal Sets?

It is very important to recognize when sets are NOT equal. Just because they look similar doesn’t mean they are the same!

1. Different Number of Elements: Set A = {1, 2, 3} and Set B = {1, 2}. These are not equal because A has 3 members and B has only 2.
2. Different Types of Numbers: Set M = {Whole numbers < 5} and Set N = {Natural numbers < 5}. M includes 0, but N does not. Therefore, M=N.
3. Different Values: Set E = {1, 2} and Set F = {3, 4}. Even though both have 2 elements, the values are different, so they are not equal.
4. Inequalities that don’t match: x<5 and x≤5. The second one includes the number 5, while the first one does not.
5. Different Formulas: y=2n and y=2n−1. Even if you use the same values for n, the results for y will be different.
6. Different Universes: If n belongs to Whole Numbers in one set and Integers in another, the set with Integers might be infinite while the other is finite.

— — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —

Parts and Important Terms

Let us break down the important terms we use when solving set problems:

1. Variable: Usually represented by x,y, or n. It is a symbol that can take different values depending on the rules of the set.
2. Natural Numbers (N): Counting numbers starting from 1 (1,2,3…).
3. Whole Numbers (W): Natural numbers including zero (0,1,2,3…).
4. Integers (I or Z): All positive and negative counting numbers and zero.
5. Condition / Inequality: A rule like x<5 or x+3<8 that tells us which variables to pick.
6. Elements: The actual members inside the curly brackets {}, such as {2, 4, 6}.
7. Cardinal Number: The total count of elements in a set, written as n(A).

— — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —

How to Solve Set Problems

To find the elements of a set given in set-builder form, we follow a step-by-step method similar to solving linear equations.

Step 1: Solve the Inequality or Equation

If the set says x+3<8, we need to find x.

• On transposing 3 to the other side, it becomes negative: x<8−3.
• On subtracting, we get x<5.
• Why we do this: We must isolate x to see the boundary of our set.

Step 2: Identify the “Universe”

Check if x is a Natural Number, Whole Number, or Integer.

• If x<5 and x is a Whole Number, we pick {0, 1, 2, 3, 4}.
• If x<5 and x is a Natural Number, we pick {1, 2, 3, 4}.

Step 3: Solve the Formula (If any)

Sometimes the set is a collection of a formula like 2n−1.

• First, find the values of n.
• On multiplying and on subtracting for each value of n, find the final elements.

— — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —

Solved Examples

Let us look at some very detailed solved examples to master this topic.

Example 1) Is the set of multiples of 8 finite or infinite? Solution —

1. The multiples of 8 are found by 8×n, where n is a natural number (1,2,3…).
2. Since natural numbers are endless, we can never finish listing the multiples (8,16,24,32,40…).
3. There is no end point. Ans. Infinite Set

Example 2) Find the elements of Set M = {x∣x∈W,x+3<8}. Solution —

1. We start with the inequality: x+3<8.
2. On transposing 3 to the RHS: x<8−3.
3. This gives x<5.
4. Since x must be a Whole Number, we list all whole numbers starting from 0 that are less than 5.
5. The elements are 0, 1, 2, 3, and 4. Ans. M = {0, 1, 2, 3, 4}

Example 3) If Set E = {x∣x2+8x−9=0} and Set F = {1, -9}, are they equal? Solution —

1. We must solve the quadratic equation x2+8x−9=0.
2. We need two numbers that multiply to -9 and add up to 8. These are 9 and -1.
3. On factoring: x2+9x−x−9=0.
4. x(x+9)−1(x+9)=0.
5. (x+9)(x−1)=0.
6. Either x+9=0 (so x=−9) or x−1=0 (so x=1).
7. So, Set E = {1, -9}.
8. Since Set F is also {1, -9}, the elements and count are identical. Ans. Yes, they are Equal Sets.

Example 4) Is the set of integers less than 10 finite or infinite? Solution —

1. Integers less than 10 include 9, 8, 7… 0, -1, -2, -3….
2. On the left side of the number line, negative integers go on forever.
3. Because there is no starting point or “left boundary,” we cannot count the total elements. Ans. Infinite Set

Example 5) Set A = {x∣x∈N,x❤} and Set B = {y∣y2−3y+2=0}. Are they equal? Solution —

1. For Set A: Natural numbers less than 3 are 1 and 2. So, A = {1, 2}.
2. For Set B: We solve y2−3y+2=0.
3. Two numbers that multiply to 2 and add to -3 are -2 and -1.
4. On factoring: y(y−2)−1(y−2)=0.
5. (y−2)(y−1)=0, which gives y=2 and y=1.
6. So, B = {1, 2}.
7. Both sets have exactly the same elements. Ans. Yes, A and B are Equal Sets

Example 6) Is the set {x∣x=3n−2,n∈W,n≤8} finite? Solution —

1. First, find values for n. Since n is a Whole Number and n≤8, n={0,1,2,3,4,5,6,7,8}.
2. There are 9 values for n.
3. For each n, we will get one value for x.
4. Since we can count the values of n, we can count the values of x. Ans. Finite Set

Example 7) Is the set {x∣x=3n−2,n∈I,n≤8} finite? Solution —

1. Here, n belongs to Integers (I).
2. Integers ≤8 include 8, 7, 6… 0, -1, -2, -3 and so on forever.
3. Since there are an uncountable number of values for n, there will be an uncountable number of values for x. Ans. Infinite Set

— — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —

Practice Questions

Try solving these on your own to test your understanding:

1. Identify if the set of multiples of 5 is Finite or Infinite.
2. If Set A = {3, 6, 9} and Set B = {3n∣n∈N,n≤3}, are they equal?
3. Find the elements of Set P = {x∣x∈W,x+2<5}.
4. Is the set of even numbers between 10 and 20 finite?
5. Set X = {x∣x2−5x+6=0}. Set Y = {2, 3}. Are they equal?
6. Is the set of all people living in India a finite set?
7. If n∈N, is the set {n1​} finite or infinite?
8. Set M = {letters in the word ‘MATHEMATICS’}. Is it finite?

— — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —

Note Section (Important Rules)

1. Check the “n”: Always check if the variable belongs to N (starts at 1), W (starts at 0), or I (includes negatives).
2. Count vs. Content: For equal sets, the number of elements must be the same, and the actual values must be the same.
3. Endless Negatives: Any set of integers that says “less than” a number without a lower limit is usually Infinite because negative numbers never end.
4. Empty Set: A set with no elements (like “all months with 40 days”) is always Finite because its count is zero.

— — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —

Conclusion

In this chapter, we have learned how to look at a collection of numbers and decide if it is Finite (countable) or Infinite (never-ending). We also learned that for two sets to be Equal, they must be identical twins — carrying the same number of members and the exact same values. By solving inequalities and quadratic equations, we can turn tricky “Set-Builder” descriptions into simple lists of numbers. Keep practicing these steps, and you will find that Sets are one of the most organized and logical parts of Mathematics!