Sets and their representations

What is a set?

In plain words: a set is a collection where, for any object you pick, there is a clear yes-or-no answer to “is it in?”. The vowels of the English alphabet form a set: a is in, b is not. “The five most renowned mathematicians of the world” is not a set, because people would disagree about who belongs (NCERT page 2). Once a collection is well defined, the order of its elements and any repetitions do not matter; only which objects are in it.

Well-defined collections

NCERT’s examples of sets include the odd natural numbers less than 10, the rivers of India, the vowels of the English alphabet, the prime factors of 210 (2, 3, 5 and 7) and the solutions of x² − 5x + 6 = 0, namely 2 and 3 (NCERT Section 1.2, page 1). In each case we can decide whether a given object belongs: the river Ganga belongs to the set of rivers of India and the Nile does not. Sets appear across subjects too: the SI base units form the set {m, kg, s, A, K, mol, cd}, with exactly seven elements.

Some sets are used so often that they have their own symbols (NCERT page 2):

Here N = {1, 2, 3, …}: 0 is not a natural number (NCERT Section 1.4, page 7).

In the set V of vowels, a ∈ V but b ∉ V; in the set P of prime factors of 30, 3 ∈ P but 15 ∉ P (NCERT page 2). The symbol ∈ is the Greek letter epsilon.

Roster form

There are two ways to write a set: the roster (or tabular) form and the set-builder form (NCERT page 2).

In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces { }.

NCERT Mathematics (Class 11), Section 1.2, page 2

For example, the even positive integers less than 7 are {2, 4, 6}, and the natural numbers that divide 42 are {1, 2, 3, 6, 7, 14, 21, 42}. Two rules go with the roster form (NCERT page 3):

• order does not matter: {1, 2, 3, 6, 7, 14, 21, 42} and {1, 3, 7, 21, 2, 6, 14, 42} are the same set (see equal sets);
• an element is not repeated: the letters of the word SCHOOL form the set {S, C, H, O, L}, with O written once.

A set with infinitely many elements can sometimes be written in roster form by listing enough elements to show the pattern, followed by three dots: the odd natural numbers are {1, 3, 5, …}. Dots between two listed elements mean the pattern continues up to the last one: {1, 4, 9, …, 100} is finite, with ten elements. Dots at both ends, as in {…, −2, −1, 0, 1, 2, …}, run on both ways (NCERT page 7).

Set-builder form

In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.

NCERT Mathematics (Class 11), Section 1.2, page 3

The set of vowels is written V = {x : x is a vowel in the English alphabet}, read “the set of all x such that x is a vowel in the English alphabet”. The braces stand for “the set of all” and the colon for “such that” (NCERT page 3). Any letter can stand for the element: x, y, z or n. Beyond the NCERT text, the set x comes from can also be written before the colon, {x ∈ Z : −3 ≤ x < 7}, and some books use a bar for the colon, {x | x is a vowel}; both mean the same.

For example, A = {x : x is a natural number and 3 < x < 10} has the elements 4, 5, 6, 7, 8 and 9.

Which form to use. The roster form is best for small sets and for showing exactly what is in a set. The set-builder form is best for large or infinite sets, and for sets such as the real numbers between 1 and 2, whose elements cannot be listed in any pattern.

Changing from one form to the other

• Set-builder to roster: find every element that has the property. {x : x is a positive integer and x² < 40} = {1, 2, 3, 4, 5, 6}, since 6² = 36 < 40 but 7² = 49 > 40 (NCERT Example 2, page 3).
• Roster to set-builder: find a property shared by every element and by nothing else. {1, 4, 9, 16, 25, …} = {x : x is the square of a natural number}, or {x : x = n², n ∈ N} (NCERT Example 3, page 4).

The solution set of an equation is a common case: x² + x − 2 = 0 factorises as (x − 1)(x + 2) = 0, so its solution set is {1, −2} (NCERT Example 1, page 3).

Solved examples

Common mistakes to avoid

Quick revision summary

• A set is a well-defined collection of objects; a ∈ A means a belongs to A, b ∉ A means it does not.
• Standard sets: N, Z, Q, R, Z⁺, Q⁺, R⁺; 0 is not in N.
• Roster form: list the elements in braces, separated by commas; order does not matter and no element is repeated.
• Set-builder form: {x : property of x}, read “the set of all x such that …”.
• Infinite sets with a pattern can be written {1, 3, 5, …}; sets without a pattern need set-builder form.

Questions students ask

What is the difference between roster form and set-builder form?

Roster form lists every element of a set inside braces, while set-builder form describes the elements by a property that they have and no other object has. The multiples of 5 from 5 to 20 are {5, 10, 15, 20} in roster form and {x : x = 5n, n ∈ N and 1 ≤ n ≤ 4} in set-builder form. Roster form shows at a glance what a small set holds; set-builder form is shorter for large sets and is needed for sets such as the real numbers between 0 and 1, which cannot be listed at all.

What are the other names for roster form and set-builder form?

Roster form is also called the tabular form or the listing method, and set-builder form is also called the rule method or the property method; NCERT names them the roster (or tabular) form and the set-builder form. Whatever the names, a set is the same in either form: {2, 3, 5, 7} in tabular form and {x : x is a prime number less than 10} in rule form are the same set. A Venn diagram is a third way to show a set, as a picture rather than in symbols.

What do N, Z, Q, R and T stand for in sets?

In NCERT’s notation N is the set of natural numbers {1, 2, 3, …}, Z the integers, Q the rational numbers, R the real numbers and T the irrational numbers, that is the real numbers that are not rational. The letter Z comes from the German word Zahlen, “numbers”, and Q from “quotient”, because every rational number is a quotient p/q of integers with q ≠ 0. They fit together as N ⊂ Z ⊂ Q ⊂ R, with T the part of R outside Q, as shown in subsets of the real numbers.

Is 0 a natural number in Class 11 sets?

No, 0 is not a natural number in NCERT Class 11 Mathematics, where N = {1, 2, 3, …}; adding 0 gives the whole numbers {0, 1, 2, 3, …}. Some university books and the ISO standard for mathematical notation do include 0 in N, so check how a source defines N before answering. The difference changes answers: {x : x ∈ N and x < 3} = {1, 2}, but the whole numbers less than 3 form {0, 1, 2}.

Is the chapter Sets in the JEE Main 2027 syllabus?

Yes, sets are in the JEE Main 2027 Mathematics syllabus published by the NTA: Unit 1, Sets, Relations and Functions, begins with “Sets and their representation”, followed by union, intersection and complement of sets and their algebraic properties, and the power set. The same Unit 1 appears in the Mathematics part of Paper 2A (B.Arch.) and Paper 2B (B.Planning). The syllabus does not list the empty, finite, equal or universal sets by name; they are the basic language of the unit, as in NCERT Chapter 1.

What is the difference between set-builder notation and interval notation?

Set-builder notation can describe any set by a property, while interval notation is a shorthand only for an unbroken stretch of the real line, such as all the real numbers between two endpoints. So {x : x ∈ R and 2 < x ≤ 6} can be written as the interval (2, 6], but {x : x ∈ Z and 2 < x ≤ 6} = {3, 4, 5, 6} cannot, because it holds only the integers in that range. The bracket rules for open, closed and half-open intervals are on the subsets page, under intervals as subsets of R.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. Let A = {1, 2, 3, 4, 5, 6}. Insert ∈ or ∉: (i) 5 … A; (ii) 8 … A; (iii) 0 … A; (iv) 4 … A; (v) 2 … A; (vi) 10 … A (NCERT Exercise 1.1, Q2).

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   (i) 5 ∈ A; (ii) 8 ∉ A; (iii) 0 ∉ A; (iv) 4 ∈ A; (v) 2 ∈ A; (vi) 10 ∉ A.

2. Q2. Write in roster form: (i) B = {x : x is a natural number less than 6}; (ii) the set of letters in BETTER (NCERT Exercise 1.1, Q3 (ii), (vi)).

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   (i) {1, 2, 3, 4, 5}; (ii) {B, E, T, R}.

3. Q3. Write the solution set of x² − x − 6 = 0 in roster form.

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   (x − 3)(x + 2) = 0, so x = 3 or x = −2: the solution set is {3, −2}.

4. Q4. List all the elements of: (i) C = {x : x is an integer and x² ≤ 4}; (ii) D = {x : x is a letter in the word LOYAL}; (iii) E = {x : x is a month of a year not having 31 days} (NCERT Exercise 1.1, Q5 (iii)–(v)).

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   (i) {−2, −1, 0, 1, 2}; (ii) {L, O, Y, A}; (iii) {February, April, June, September, November}.

5. Q5. Write {2, 4, 6, …} in set-builder form (NCERT Exercise 1.1, Q4 (iv)).

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   {x : x is an even natural number}, or {x : x = 2n, n ∈ N}.

6. Q6. Match (i) {1, 2, 3, 6}, (ii) {2, 3}, (iii) {M, A, T, H, E, I, C, S}, (iv) {1, 3, 5, 7, 9} with (a) {x : x is a prime number and a divisor of 6}, (b) {x : x is an odd natural number less than 10}, (c) {x : x is a natural number and a divisor of 6}, (d) {x : x is a letter of the word MATHEMATICS} (NCERT Exercise 1.1, Q6).

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   (i)–(c), (ii)–(a), (iii)–(d), (iv)–(b).