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Class 11 | CBSE | MATHEMATICS | SETS
CHAPTER-1 | SET | SET AND THEIR REPRESENTATIONS

MATHEMATICS CLASS 11 :

Class 11 Mathematics – Chapter 1: SETS (Sections 1.1 & 1.2), written strictly from a teacher’s perspective, keeping CBSE board expectations and concept clarity in mind.

CHAPTER 1 – SETS

(Sections 1.1 & 1.2)

Class XI – Mathematics (CBSE)

1.1 INTRODUCTION TO SETS

The concept of a set is one of the most fundamental ideas in mathematics. Almost every branch of mathematics—such as algebra, geometry, probability, relations, functions, sequences, and statistics—is based on set theory.

👉 Without understanding sets, higher mathematics becomes difficult.

🔹 Historical Background

The theory of sets was developed by the German mathematician Georg Cantor (1845–1918).
He introduced sets while studying trigonometric series, and his work laid the foundation of modern mathematics.

🔹 Why are Sets Important?

Sets help us to:


1.2 SETS AND THEIR REPRESENTATIONS


What is a Set?

In mathematics, a set is defined as:

A well-defined collection of objects.

✔ Well-defined means

We must be able to clearly decide whether an object belongs to the collection or not.


Examples of Well-Defined Collections

✔ In each case, there is no ambiguity.


❌ Not a Well-Defined Collection

Reason:
The term “most renowned” is subjective and varies from person to person.


Standard Sets Used in Mathematics

SymbolMeaning
NSet of all natural numbers
ZSet of all integers
QSet of all rational numbers
RSet of all real numbers
Z⁺Set of positive integers
Q⁺Set of positive rational numbers
R⁺Set of positive real numbers

These symbols are frequently used in exams.


Important Terminology


Membership Symbols

SymbolMeaning
∈belongs to
∉does not belong to

Examples:


Representation of Sets

There are two standard methods:

1. Roster (Tabular) Form

2. Set-Builder Form


1️⃣ ROSTER (TABULAR) FORM

📌 Definition

In this method:


Examples

  1. Even positive integers less than 7

    \[ \{2,\;4,\;6\} \]

  2. Natural numbers dividing 42

    \[ \{1,\;2,\;3,\;6,\;7,\;14,\;21,\;42\} \]

✔ Order does not matter

\[ \{1,\;3,\;7,\;21,\;2,\;6,\;14,\;42\} \] \[ \text{is the same set} \]

  1. Vowels in English alphabet

    \[ \{a,\;e,\;i,\;o,\;u\} \]

  2. Odd natural numbers

    \[ \{1,\;3,\;5,\;\ldots\} \]

(The dots indicate continuation.)


⚠ Important Notes (Very Important for Exams)

Example:

Letters of the word SCHOOL

\[ \{S,\;C,\;H,\;O,\;L\} \]

(O is written only once.)


2️⃣ SET-BUILDER FORM

📌 Definition

In this method:


General Form

\[ \{x : \text{property of } x\} \]

Read as:

“The set of all x such that …”


Examples

  1. Vowels in English alphabet

    \[ V = \{x : x \text{ is a vowel in the English alphabet}\} \]

  2. Natural numbers between 3 and 10

    \[ A = \{x : x \in N\} \] \[ 3 < x < 10 \]

✔ Elements:

\[ \{4,\;5,\;6,\;7,\;8,\;9\} \]

Examples (Sets)

Example 1: Write the set of natural numbers less than 6 in roster form.
\[ \text{Given condition: } x \in N \text{ and } x < 6 \] \[ \text{Natural numbers less than 6 are:} \] \[ \{1,\;2,\;3,\;4,\;5\} \]
Example 2: Write the set of integers between −2 and 3 in roster form.
\[ \text{Given: } x \in Z \] \[ -2 \le x \le 3 \] \[ \text{So the required set is:} \] \[ \{-2,\;-1,\;0,\;1,\;2,\;3\} \]
Example 3: Write the set {1, 4, 9, 16} in set-builder form.
\[ \text{Observe that:} \] \[ 1 = 1^2,\quad 4 = 2^2,\quad 9 = 3^2,\quad 16 = 4^2 \] \[ \text{Hence, the set-builder form is:} \] \[ \{x : x = n^2,\; n \in N,\; 1 \le n \le 4\} \]

Converting Roster Form to Set-Builder Form

Roster FormSet-Builder Form
{1, 2, 3, 6, 7, 14, 21, 42}{x : x is a natural number which divides 42}
{a, e, i, o, u}{x : x is a vowel in English alphabet}
{1, 3, 5, …}{x : x is an odd natural number}

📌 EXAM TIPS (VERY IMPORTANT)

✔ Always check whether a collection is well-defined
✔ Use correct symbols (∈, ∉, :, { })
✔ Avoid repetition of elements
✔ Order does not matter in sets
✔ Practice conversion between roster and set-builder forms


SUMMARY


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