Exercise 2.4 — Finite and Infinite Sets

What are Finite and Infinite Sets?

In Class 10 Mathematics, the chapter on Sets introduces the idea of classifying sets based on the number of elements they contain. A set is called a finite set if its elements can be counted completely, meaning there is a clear starting point and a clear ending point. For example, the set of natural numbers less than 100, or the set of students in a classroom, are finite sets because we can list every element and stop at a definite point.

On the other hand, a set is called an infinite set if it is not possible to count all its elements because there is no fixed ending point. The set of all natural numbers, the set of all multiples of 15, or the set of all straight lines passing through a single point are examples of infinite sets. Interestingly, the null set (or empty set) is always considered a finite set because it has zero elements, which is a countable quantity.

Cardinal Number of a Set

The total number of elements present in a finite set is called its cardinal number, written as n(A) for a set A. This concept helps in quickly describing the size of a set without listing every element each time.

• n(R) = 4 — if R = {a, b, c, d}, it has 4 elements
• n(A) = 5 — the set of vowels in the English alphabet has 5 elements
• n(M) = 3 — the set of letters in the word "MAY" has 3 elements
• n(S) = 99 — the set {1, 2, 3, ..., 98, 99} has 99 elements
• n(φ) = 0 — the cardinal number of the null set is always zero

Identifying Empty Sets

An empty set (also called a null set) is a set that contains no elements at all, and it is denoted by the symbol φ or { }. Many problems in this exercise ask students to determine whether a given set is empty by checking if any element actually satisfies the given condition.

For instance, the set of integers lying strictly between 2 and 3 is empty, because there is no integer in that range. Similarly, the set of natural numbers smaller than 1 is empty, since natural numbers start from 1. Another example is the set of odd numbers that leave a remainder of zero when divided by 2 — this is also empty, because by definition, odd numbers always leave a remainder of 1 when divided by 2, never zero.

Worked Example: Finite or Infinite?

Consider the set C = {1/2, 2/2, 3/2, ...}. To decide whether this set is finite or infinite, look for an ending element. Since the pattern continues forever with no last term, this set is infinite. In contrast, the set D = {1, 2, 3, 4} clearly has only four elements with a definite end, so it is finite.

Another useful example is the set {x : x is a day of the week}. Even though this set is described using a condition rather than a list, it is finite because there are exactly seven days in a week, and all of them can be listed: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, and Sunday.

Common Mistakes to Avoid

• Confusing the null set with the number zero — the null set is a set with no elements, while zero is a number
• Assuming a set described by a condition is automatically infinite without checking if elements actually exist
• Forgetting that the null set is itself classified as finite, with cardinal number 0
• Mixing up "no solution" (empty set) with "infinite solutions" when solving equations to identify a set
• Not checking both upper and lower bounds when a set is defined using inequalities, such as x < 5 and x > 7

Practice with Real-Life Sets

To build confidence, try classifying everyday examples as finite, infinite, or empty. The set of months in a year is finite, with exactly 12 elements. The set of prime numbers smaller than 99 is also finite, since prime numbers can be listed and counted within that range. However, the set of all circles passing through the origin (0, 0) is infinite, because infinitely many circles of different radii can pass through a single point.

Similarly, the set of all lines parallel to the X-axis is infinite, since lines can be drawn at every possible height. Practicing such classification questions strengthens conceptual clarity, which is essential for objective-type questions in CBSE, Telangana, and Andhra Pradesh board exams.

What This Lesson Prepares You For

Understanding finite, infinite, empty sets, and cardinal numbers builds the foundation needed for set operations such as union, intersection, and complement, which appear in the next sections of this chapter. A solid grasp here also helps with topics involving counting and probability later in the syllabus.

To continue building your understanding of sets, revisit the basics in Exercise 2.3 on equal sets and subsets, and explore how set notation connects to real numbers in the next chapter.