Exercise 2.3 — Equal Sets

Understanding Equal Sets

In Class 10 Mathematics, the chapter on Sets introduces an important idea called equal sets. Two sets A and B are called equal sets if every element of A is also an element of B, and every element of B is also an element of A. In simple terms, both sets contain exactly the same members, regardless of the order in which they are written. If A and B are equal, we write A = B; if not, we write A ≠ B.

This concept connects directly with the idea of subsets. Two sets are equal if and only if A is a subset of B and B is a subset of A at the same time. This relationship is one of the most frequently tested concepts in CBSE, Telangana, and Andhra Pradesh board exams, so students should be confident in identifying equal sets using both roster form and set-builder form.

Key Concepts from Exercise 2.3

Exercise 2.3 builds on the definition of equal sets and helps students practice converting sets written in set-builder form into roster form before comparing them. This step is crucial because two sets that look different in set-builder notation can actually represent the same collection of elements once written out.

• Equal Sets — sets containing exactly the same elements, written as A = B
• Roster Form — a set written by listing all its elements, such as {a, b, c}
• Set-Builder Form — a set described using a rule or property, such as {x : x is a vowel}
• Subset Condition — A = B only when A ⊂ B and B ⊂ A both hold true
• Listing Subsets — for a set with n elements, the total number of subsets is 2 to the power n

Worked Example: Checking for Equal Sets

Consider the sets formed from the letters of three words: A from "FOLLOW", B from "FLOW", and C from "WOLF". Writing each in roster form, A becomes {F, O, L, W}, B becomes {F, L, O, W}, and C becomes {W, O, L, F}. Although the letters are written in a different order in each set, all three sets contain exactly the same four letters: F, O, L, and W.

Since every element of A appears in B and every element of B appears in A, we conclude A = B. The same logic applies to B and C, and to A and C. Therefore, all three sets A, B, and C are equal. This example teaches an important lesson: the order of elements in a set never matters, and repeated letters in a word are written only once when forming a set.

Comparing Sets Written in Different Forms

Many questions in this exercise ask students to decide whether A = B or A ≠ B when one set is given in roster form and the other in set-builder form. For example, the set of positive even integers less than 10 written in roster form is {2, 4, 6, 8}, which has only four elements. If this is compared with {2, 4, 6, 8, 10}, the two sets are not equal because 10 is present in one set but not the other.

Similarly, a set described as "multiples of 10" includes {10, 20, 30, 40, ...}, while a set listed as {10, 15, 20, 25, 30, ...} includes numbers like 15 and 25 that are not multiples of 10. Even though both sets share some common numbers, they are not identical, so A ≠ B. Careful conversion to roster form before comparing is the safest way to avoid mistakes in such problems.

Common Mistakes to Avoid

• Assuming two sets are equal just because they share some common elements — equality requires every element to match exactly
• Forgetting to convert set-builder notation into roster form before comparing sets
• Mixing up the symbols ⊂ (subset) and = (equal) — equal sets satisfy the subset condition in both directions
• Missing or duplicating elements while listing subsets of a set
• Forgetting to include the empty set { } and the original set itself while listing all subsets

Listing All Subsets of a Set

Another important skill covered in this exercise is listing all subsets of a given set. A subset is any set whose elements are all contained within another set, including the empty set and the set itself. For a set with 2 elements, there are 4 subsets; for a set with 3 elements, there are 8 subsets; and for a set with 4 elements, there are 16 subsets. This pattern follows the formula 2 raised to the power of the number of elements in the set.

For example, for the set B = {p, q}, the subsets are { }, {p}, {q}, and {p, q}. Practicing this systematically for sets with 3 and 4 elements helps students build accuracy and speed, which is especially useful for board exam questions that ask for the total number of subsets or proper subsets of a given set.

What This Lesson Prepares You For

A strong grasp of equal sets and subsets lays the foundation for more advanced set operations such as union, intersection, and difference of sets, which are covered in later exercises of this chapter. Students preparing for CBSE, Telangana, and Andhra Pradesh board exams should also revisit the basics of set notation before moving forward.

To strengthen your understanding, explore related topics such as the introduction to real numbers and polynomials, which often use set notation to describe number systems and solution sets.