Concepts & Solved Examples
Learn definitions, roster vs. set-builder representations, empty sets, equality, and Georg Cantor's history.
Exercise 1.1 Solutions
Detailed step-by-step solutions to all parts of Exercise 1.1 with interactive answer details.
Chapter Worksheets
Three levels (Easy, Medium, Hard) of custom worksheets designed to perfect your set theory skills.
Revision Notes
Quick formulas, notation keys, common mistakes, and key takeaways for exam revision.
Interactive Tests
Basic, Standard, and Advanced quizzes with 10 questions each to test your understanding of sets basics.
What You'll Learn in This Section
Sets are the fundamental building blocks of modern mathematical language. In this chapter, we lay the groundwork for set theory, transitioning from everyday informal groupings to rigorous mathematical notation.
• Definition of a Set: Understanding well-defined collections (distinguishing sets from subjective groupings).
• Set Representations: Writing sets in Roster (Tabular) Form vs. Set-Builder Description Form.
• Elements & Membership: Using the symbols $\in$ (belongs to) and $\notin$ (does not belong to).
• Types of Sets: Identifying Finite, Infinite, Empty (Null/Void), and Equal Sets.
• Historical Context: Georg Cantor's pioneering discoveries in set theory and transfinite math.
Start Learning
Click on Concepts & Solved Examples to begin studying the theory, or go straight to Exercise 1.1 Solutions to check your homework answers.
Set Theory: Core Concepts & Solved Examples
What is a Set?
We often talk about groups of things in our daily lives, like different blood groups ($A, B, AB, O$), a set of books in algebra, or a set of students participating in a quiz contest. In mathematics, we formalize this idea.
A set is a well-defined collection of objects. The objects in a set are called the elements or members of the set.
A collection is said to be well-defined if there is no confusion or ambiguity in deciding whether any given object belongs to the collection or not.
Well-Defined (Sets)
- The set of vowels in the English alphabet: $\{a, e, i, o, u\}$
- The set of girls participating in a Kho-Kho match
- Whole numbers less than or equal to 5
Not Well-Defined (Not Sets)
- "The three best students of a class" (The word "best" is subjective).
- "The 3 most interesting books" (Interest depends entirely on the reader).
Representations of a Set
There are two primary methods used to write and represent sets:
1. Roster Form (Tabular Form)
In this method, we list all the elements of the set inside curly braces, separating them with commas. Sets are denoted by capital letters.
- The set of vowels in English alphabet: $V = \{a, e, i, o, u\}$
- If an element $e$ belongs to the set $V$, we write: $e \in V$ (where the symbol $\in$ means "belongs to").
- If an element $c$ does not belong to the set $V$, we write: $c \notin V$ (where the symbol $\notin$ means "does not belong to").
(i) Set of whole numbers less than or equal to 5.
(ii) Set of first 4 terms of the A.P., whose first term is $-3$ and common difference is $4$.
Show Solution
$\Rightarrow A = \{0, 1, 2, 3, 4, 5\}$
(ii) Let the set be $B$. The first term is $-3$. Common difference is $4$. The terms are:
$t_1 = -3$, $t_2 = -3 + 4 = 1$, $t_3 = 1 + 4 = 5$, $t_4 = 5 + 4 = 9$.
$\Rightarrow B = \{-3, 1, 5, 9\}$
2. Set-Builder Form
In this method, instead of listing the elements, we specify them by describing their common property. All elements in the set must share a property which no element outside the set possesses.
For example, the set of vowels $V = \{a, e, i, o, u\}$ is written as:
This is read as: "The set of all elements $x$ such that $x$ is a vowel of the English alphabet."
(i) $\{1, -1\}$
(ii) $\left\{\frac{2}{3}\right\}$
(iii) $\left\{\frac{1}{2}, \frac{1}{4}, 1, 2\right\}$
Show Solution
$\{x \mid x \text{ is an integer and } x^2 = 1\}$
(ii) The fraction $\frac{2}{3} = 0.666... \approx 0.\overline{6}$. We can express this as:
$\{x \mid x \text{ is a fraction equivalent to } 0.\overline{6}\}$
(iii) The elements can be written as powers of 2, i.e., $2^{-1}, 2^{-2}, 2^0, 2^1$. Alternatively, they can be described as:
$\{x \mid x = 2^n \text{ where } n \in \mathbb{Z} \text{ and } -2 \le n \le 1\}$
The theory of sets was developed by German mathematician Georg Cantor, the founder of set theory. He studied at Zurich and Berlin. Cantor received his doctorate in 1867 in Number Theory. His most famous discovery is that the set of real numbers is uncountable (having a larger cardinality than integers).
Finite and Infinite Sets
Sets can contain a limited number of elements, or their elements can go on forever.
- Finite Set: A set containing a countable (finite) number of elements. E.g., the set of students in your class.
- Infinite Set: A set whose elements cannot be listed completely because they are endless. E.g., the set of prime numbers.
(i) $\{x \mid x \text{ is an integer lying between } 5 \text{ and } 91\}$
(ii) Set of coordinates of the points lying on a unit circle.
(iii) $\{x \mid x \in \mathbb{N} \text{ and } (x, y) \text{ lies on the line } 2x + y = 8\}$
(iv) $\{x \mid x \text{ is the number of animals on earth}\}$
(v) $\{x \mid x \text{ is a digit in the decimal expansion of } \sqrt{2}\}$
Show Solution
(ii) Infinite: A circle consists of infinitely many points, so there are infinite coordinates.
(iii) Finite: Since $x \in \mathbb{N}$, the only possible natural values of $x$ that satisfy $2x + y = 8$ for a positive $y$ are $1, 2, 3$. Hence it is finite.
(iv) Finite: Although extremely large, the total count of animals on Earth is a definite, finite number.
(v) Finite: The decimal expansion of $\sqrt{2} = 1.41421356...$ is non-terminating, but the set of digits occurring in it is subset of $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$, which has at most 10 elements. Thus it is finite.
(i) $\{x \mid x \text{ is an odd natural number between } 3 \text{ (excluding) and } 11 \text{ (including)}\}$
(ii) $\{x^2 \mid x \text{ is a whole number less than or equal to } 3\}$
Show Solution
$\Rightarrow \{5, 7, 9, 11\}$
(ii) The whole numbers less than or equal to $3$ are $0, 1, 2, 3$. Their squares are $0^2=0$, $1^2=1$, $2^2=4$, $3^2=9$.
$\Rightarrow \{0, 1, 4, 9\}$
(i) Real numbers between 2 and 5.
(ii) Fractions whose numerator and denominator are natural numbers and denominator exceeds the numerator by 1.
Show Solution
Roster Form: Not possible. Because we cannot list all the infinite real numbers between 2 and 5, nor do we have a pattern to list consecutive elements.
Set Builder Form: $\{x \mid x \in \mathbb{R} \text{ and } 2 < x < 5\}$
(ii) Fractions with denominator = numerator + 1:
Roster Form: $\left\{\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \dots\right\}$
Set Builder Form: $\left\{x \mid x = \frac{n}{n+1} \text{ where } n \in \mathbb{N}\right\}$
1. Order of elements is NOT important: $\{1, 2, 3\} = \{3, 2, 1\} = \{2, 3, 1\}$.
2. Repetition of elements is NOT allowed: A set contains only distinct elements. For example, the set of elements in "BANANA" is $\{B, A, N\}$ (only 3 elements, not 6).
(i) $y$ ... $\{a, x, y, z\}$
(ii) $4$ ... $\{1, 2, 5, 7, 11\}$
(iii) $2$ ... $\dots \emptyset$
(iv) $5$ ... $\{x \mid x \text{ is a natural number } \ge 5\}$
Show Solution
(ii) $4 \notin \{1, 2, 5, 7, 11\}$ ($4$ is not in the set)
(iii) $2 \notin \emptyset$ (The empty set has no elements)
(iv) $5 \in \{x \mid x \text{ is a natural number } \ge 5\}$ ($5$ is greater than or equal to $5$)
(i) $\{2, 3, 5, 7, 11\}$
(ii) $\{1, 3, 5, 7, 9, 11\}$
Show Solution
(ii) Set Builder Form: $\{x \mid x \text{ is an odd natural number less than } 13\}$
Empty or Null Set
Consider the set $\{x \mid x \text{ is a prime number which is composite}\}$. Since no prime number can be composite, this set contains no elements at all.
A set which does not contain any element is called an empty, null, or void set. It is denoted by the Greek symbol $\emptyset$ (phi) or empty braces $\{\}$.
(i) Set of even prime numbers.
(ii) Set of composite numbers having at most 2 factors.
(iii) Set of numbers which are both rational and irrational.
(iv) Set of irrational numbers whose decimal expansion terminates.
Show Solution
(ii) Empty Set: Composite numbers must have more than 2 factors. Thus, no such number exists.
(iii) Empty Set: Real numbers are partitioned into rationals and irrationals; no number can be both.
(iv) Empty Set: By definition, the decimal expansion of an irrational number is non-terminating and non-recurring.
Equality of Sets
Two sets $A$ and $B$ are said to be equal ($A = B$) if they have exactly the same elements. If they do not, we write $A \neq B$.
Equal Sets Example
• $\{2, 5, -1, 0\} = \{0, -1, 5, 2\}$ (order doesn't matter)
• $\{1, 2, 2, 2, 7, 7, 7\} = \{1, 2, 7\}$ (repetitions don't add elements)
Unequal Sets Example
• $\{1, 2, 3\} \neq \{1, 2\}$
• $\{x \mid x^2 = 4\} = \{2, -2\} \neq \{2\}$
Exercise 1.1 — Ideal Textbook Solutions
Question 1
List the elements of the following sets
(a) $\{x \mid x \text{ is an integer and } x^2 = 9\}$
(b) $\{x \mid x \text{ is a positive integer less than } 5\}$
(c) $\{x \mid x \text{ is even natural number divisible by } 5\}$
(d) $\{x \mid x \in \mathbb{N} \text{ and } x < -1\}$
View Complete Solution & Explanation
(a) Solution:
The equation is $x^2 = 9$. Taking square roots on both sides, we get $x = \pm 3$.
Since both $3$ and $-3$ are integers ($\in \mathbb{Z}$), the elements are:
$\{-3, 3\}$
(b) Solution:
Positive integers are $\{1, 2, 3, \dots\}$. The ones strictly less than $5$ are:
$\{1, 2, 3, 4\}$
(c) Solution:
Even natural numbers are $\{2, 4, 6, 8, 10, \dots\}$. The ones divisible by $5$ are multiples of $10$ ($2 \times 5 = 10$).
$\{10, 20, 30, 40, \dots\}$
(d) Solution:
Natural numbers ($\mathbb{N}$) start from $1$ (i.e. $\{1, 2, 3, \dots\}$). No natural number can be negative or less than $-1$.
Therefore, this set is empty:
$\{\}$ or $\emptyset$
Question 2
Classification of Numbers
Determine which elements of the set $$A = \left\{5, -3, \frac{1}{2}, 0, \frac{2}{5}, -\frac{1}{3}, 13.4, \frac{19}{2}\right\}$$ are:
(a) Natural numbers (b) Whole numbers (c) Integers (d) Rational numbers (e) Real numbers
View Complete Solution & Explanation
Let's evaluate each number in $A$:
- $5$: Natural, Whole, Integer, Rational, Real
- $-3$: Integer, Rational, Real
- $\frac{1}{2}$: Rational, Real
- $0$: Whole, Integer, Rational, Real
- $\frac{2}{5}$: Rational, Real
- $-\frac{1}{3}$: Rational, Real
- $13.4$ (which is $\frac{134}{10} = \frac{67}{5}$): Rational, Real
- $\frac{19}{2}$: Rational, Real
Final Answers:
(a) Natural Numbers: $\{5\}$
(b) Whole Numbers: $\{0, 5\}$
(c) Integers: $\{-3, 0, 5\}$
(d) Rational Numbers: All elements in $A$ can be expressed in $\frac{p}{q}$ form. So the answer is:
$\left\{5, -3, \frac{1}{2}, 0, \frac{2}{5}, -\frac{1}{3}, 13.4, \frac{19}{2}\right\}$
(e) Real Numbers: All elements are real numbers. So the answer is:
$\left\{5, -3, \frac{1}{2}, 0, \frac{2}{5}, -\frac{1}{3}, 13.4, \frac{19}{2}\right\}$
Question 3
Write in Roster Form
(a) $\{x \mid x \text{ is a two digit number and the sum of the digits is } 5\}$
(b) $\{x \mid x \text{ is an integer and } |x| \le 9\}$
(c) $\{x \mid x \text{ is letter of the word } \text{"SWEET"}\}$
(d) $\left\{x \mid x = \frac{1}{n} \text{ where } n \text{ is a natural number and } n < 6\right\}$
(e) $\{x \mid x \text{ is a composite number}\}$
View Complete Solution & Explanation
(a) Solution:
Let $x = 10d_1 + d_2$, where $d_1, d_2$ are digits and $d_1 \ge 1$. The sum of digits is $d_1 + d_2 = 5$.
Pairs $(d_1, d_2)$ are $(1, 4), (2, 3), (3, 2), (4, 1), (5, 0)$.
$\{14, 23, 32, 41, 50\}$
(b) Solution:
Since $|x| \le 9$ and $x$ is an integer, $x$ ranges from $-9$ to $9$ inclusive.
$\{-9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$
(c) Solution:
The distinct letters in the word "SWEET" are S, W, E, T. Repetition of letters is not allowed in a set.
$\{\text{S, W, E, T}\}$
(d) Solution:
$n \in \mathbb{N}$ and $n < 6 \Rightarrow n \in \{1, 2, 3, 4, 5\}$. Substituting $n$ in $\frac{1}{n}$:
$\left\{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}\right\}$
(e) Solution:
A composite number is a positive integer greater than 1 that has a positive divisor other than 1 and itself.
$\{4, 6, 8, 9, 10, 12, 14, 15, \dots\}$
Question 4
Write in Set-Builder Form
(i) $\{2, 4, 6, 8, \dots\}$
(ii) $\{3, 6, 9, 12, 15\}$
(iii) $\{1, 4, 9, 16, \dots\}$
(iv) $\{8, 9, 10, 11, \dots\}$
(v) $\{1, 2, 3, 6\}$
View Complete Solution & Explanation
(i) Solution: These are even natural numbers.
$\{x \mid x \text{ is an even natural number}\}$ or $\{x \mid x = 2n, n \in \mathbb{N}\}$
(ii) Solution: These are the first 5 multiples of 3.
$\{x \mid x = 3n \text{ where } n \in \mathbb{N} \text{ and } 1 \le n \le 5\}$
(iii) Solution: These are squares of natural numbers ($1^2, 2^2, 3^2, 4^2, \dots$).
$\{x \mid x = n^2 \text{ where } n \in \mathbb{N}\}$
(iv) Solution: Natural numbers starting from 8.
$\{x \mid x \in \mathbb{N} \text{ and } x \ge 8\}$
(v) Solution: These numbers are divisors (or factors) of 6.
$\{x \mid x \text{ is a positive factor of } 6\}$
Question 5
Equality of Sets
Which of the following pairs of sets are equal:
(i) $\{D, E, C, E, N, T\}$ and $\{C, E, N, T, D\}$
(ii) $\{a, b, \alpha, 2\}$ and $\{a, b, 2, \alpha\}$
(iii) $\{x \mid x \text{ is zero of the polynomial } x^2\}$ and $\{x \mid x \text{ is the root of the equation } x^2 = 0\}$
(iv) $\{x \mid x \text{ has numerical value less than or equal to } 1\}$ and $\{x \mid x \text{ is the root of the equation } x^2 - 1 = 0\}$
(v) $\{5, 10, 15, 20\}$ and $\{5, 10, 15, 20, \dots\}$
(vi) $\emptyset$ and $\{\}$
View Complete Solution & Explanation
(i) Equal:
Removing duplicates from $\{D, E, C, E, N, T\}$ gives $\{C, D, E, N, T\}$. The second set is $\{C, E, N, T, D\}$, which contains the exact same elements. Since order and repetition do not change a set, they are equal.
(ii) Equal:
Both sets have elements $\{a, b, 2, \alpha\}$. Order of elements is irrelevant, so they are equal.
(iii) Equal:
• Zeros of $x^2$: only $x = 0$. So the set is $\{0\}$.
• Roots of $x^2 = 0$: only $x = 0$. So the set is $\{0\}$.
Both sets are $\{0\}$, so they are equal.
(iv) Not Equal:
• First set $\{x \mid x \text{ has numerical value } \le 1\}$ typically refers to real numbers in $[-1, 1]$, which is an infinite set.
• Second set contains roots of $x^2 - 1 = 0$, which are $x = \pm 1$. So the set is $\{-1, 1\}$.
The sets are not equal.
(v) Not Equal:
The first set has exactly 4 elements: $\{5, 10, 15, 20\}$. The second set has infinitely many elements: $\{5, 10, 15, 20, 25, \dots\}$. They are not equal.
(vi) Equal:
$\emptyset$ and $\{\}$ are two notations representing the exact same empty set (a set containing zero elements).
Question 6
Finite vs. Infinite Sets
State which of the following sets are finite or infinite:
(i) $\{x \mid x \in \mathbb{Z} \text{ and } (x - 1)(x + 2)(x - 3) = 0\}$
(ii) $\{x \mid x \text{ and } 2 \text{ are coprime}\}$
(iii) $\{x \mid x \text{ is a rational number between } 3 \text{ and } 4\}$
(iv) $\{x \mid x \text{ is an integer and } |x| < 5\}$
View Complete Solution & Explanation
(i) Finite:
The roots of the equation are $x = 1, -2, 3$. Since all three are integers, the set in roster form is $\{1, -2, 3\}$, which has exactly 3 elements. Thus, it is a finite set.
(ii) Infinite:
Two integers are coprime if their only common divisor is 1. Any odd integer is coprime to 2 (e.g., $3, 5, 7, 9, \dots$). Since there are infinitely many odd integers, the set is infinite.
(iii) Infinite:
Between any two distinct rational numbers (like 3 and 4), there lie infinitely many other rational numbers. Therefore, this set is infinite.
(iv) Finite:
The integers satisfying $|x| < 5$ are $\{-4, -3, -2, -1, 0, 1, 2, 3, 4\}$. This set has exactly 9 elements, so it is finite.
Chapter Worksheets — Sets basics
Conceptual Foundations & Representations
Perfect for testing basic definitions, element symbol usage ($\in, \notin$), and straightforward roster listings.
-
1
Explain why "the set of five intelligent scientists in India" is not a mathematically valid set, and rewrite it as a well-defined set.
-
2
If $P = \{1, 3, 5, 7, 9\}$, insert the correct symbol $\in$ or $\notin$ in the following blank spaces:
(a) $3$ ___ $P$ (b) $4$ ___ $P$ (c) $9$ ___ $P$ (d) $0$ ___ $P$ -
3
Write the set of letters in the word "COMMITTEE" in Roster form. What is the number of elements?
-
4
Represent the set of all natural numbers less than 20 that are divisible by 3 in Roster form.
Conversions & Set Properties
Focuses on roster to set-builder conversions, working with algebraic expressions, and identifying empty sets.
-
5
Write the following set in set-builder form:
$T = \left\{\frac{1}{2}, \frac{2}{5}, \frac{3}{10}, \frac{4}{17}, \frac{5}{26}\right\}$ -
6
Write the set $D = \{x \mid x \text{ is an integer and } x^2 - x - 6 = 0\}$ in Roster Form.
-
7
Determine if the following set is empty, finite, or infinite:
$X = \{x \mid x \in \mathbb{N} \text{ and } (x - 1)(x - 2) < 0\}$ -
8
Find if the sets $A = \{x \mid x^2 = 25\}$ and $B = \{5\}$ are equal, and justify your answer.
Rigorous Notations & Edge Cases
Designed for students looking for deeper conceptual challenges, nested sets, and mathematical proofs.
-
9
Let $S = \{\emptyset, 1, \{2, 3\}\}$. Determine which of the following statements are True or False:
(a) $\emptyset \in S$ (b) $\{2, 3\} \in S$ (c) $2 \in S$ (d) $\{\emptyset\} \in S$ -
10
Write the set $\{x \mid x \in \mathbb{R} \text{ and } x^2 + 1 = 0\}$ in roster form. Which type of set is this?
-
11
Is the set of all rational numbers between $\sqrt{2}$ and $\sqrt{3}$ finite or infinite? Justify.
-
12
If a set $A$ contains $n$ elements, can you write the set of all its subsets in Roster form? Describe it in Set-builder form.
Quick Revision Notes — Set basics
60-Second Summary
The core takeaways in under a minute
Set theory, established by Georg Cantor, is the math of collections. A set must be a well-defined collection of distinct elements. You represent sets using Roster Form (listing elements like $\{1, 2, 3\}$) or Set-Builder Form (describing the common property like $\{x \mid x \text{ is an integer } > 0\}$). Sets are classified as finite (countable items), infinite (endless), empty (containing no elements, denoted $\emptyset$ or $\{\}$), or equal (sharing identical elements, disregarding order and duplication).
Key Terminology & Symbols
Standard mathematical set vocabulary
| Symbol / Term | Meaning | Example |
|---|---|---|
| $\in$ | Belongs to (is an element of) | $3 \in \{1, 3, 5\}$ |
| $\notin$ | Does not belong to | $2 \notin \{1, 3, 5\}$ |
| $\emptyset$ or $\{\}$ | Empty set (Null or Void set) | $\{x \mid x \text{ is prime and composite}\}$ |
| Roster Form | Listing all elements inside braces, separated by commas | $A = \{2, 4, 6\}$ |
| Set-Builder Form | Describing elements using their common qualifying property | $A = \{x \mid x \text{ is an even natural number } < 8\}$ |
| $\mathbb{N}, \mathbb{W}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}$ | Standard sets of Natural, Whole, Integers, Rationals, Reals | $-5 \in \mathbb{Z}$, but $-5 \notin \mathbb{N}$ |
Common Mistakes to Avoid
Watch out for these classic exam traps!
$\emptyset$ is the empty set (contains 0 elements). $\{\emptyset\}$ is a set containing the empty set (contains 1 element). They are NOT the same!
Never write repeated elements in roster form. E.g., the set of letters in "HELLO" must be written as $\{H, E, L, O\}$, not $\{H, E, L, L, O\}$.
If a property is subjective (e.g. "nice people", "tall towers", "best books"), it is not well-defined and cannot form a set.
Set Theory Chapter Tests
Take a Practice Test
Choose a difficulty level to test your understanding of Set Basics & Representation. Each test is out of 24 Marks and contains 10 structured questions: 3 MCQs, 1 Assertion-Reasoning (AR), and 6 descriptive/subjective questions of varying marks. Compare your answers with official guidelines to self-grade descriptive parts.
Basic Test
Focuses on set definition, roster form listings, element membership, and equal set basics.
Standard Test
NCERT/textbook level conversions, infinite set categorization, and formal equality proofs.
Advanced Test
Complex notations, nested sets, algebraic condition sets, and Cantor's uncountability theory.