Comprehensive guide and practice questions for CBSE Class 11 Applied Mathematics curriculum (2026-27).
Comprehensive Concepts & Principles: Types of Sets
In set theory and CBSE Class 11 Applied Mathematics, sets are systematically classified according to the number and nature of elements they contain. Mastering these classifications is essential for analyzing categorical data, constructing database domain constraints, and evaluating logical boundary conditions.
Core Taxonomy of Sets:
1. Empty Set (Null Set / Void Set): A set that contains no elements at all, denoted by the Greek symbol $\emptyset$ or empty braces $\{\}$.
• Cardinality: $n(\emptyset) = 0$.
• Critical Notation Alert: The set $\{\emptyset\}$ or $\{0\}$ is NOT an empty set; it is a singleton set containing one element!
2. Singleton Set (Unit Set): A set containing exactly one element ($n(A) = 1$).
• Example: $S = \{x : x \in \mathbb{N}, x^2 = 4\} = \{2\}$.
3. Finite Set: A set that has a definite, countable number of elements (its cardinality $n(A) \in \mathbb{W}$).
• Example: The set of days in a week ($n = 7$).
4. Infinite Set: A set whose elements cannot be counted to a terminal end.
• Examples: The set of all natural numbers $\mathbb{N}$, the set of all real numbers between 0 and 1, the set of points on a line segment.
5. Equal Sets ($A = B$): Two sets $A$ and $B$ are equal if and only if they possess identical elements:
$$A = B \iff (x \in A \iff x \in B)$$
6. Equivalent Sets ($A \approx B$): Two finite sets are equivalent if they have the same cardinal number of elements:
$$A \approx B \iff n(A) = n(B)$$
• Golden Rule: All equal sets are equivalent, but equivalent sets are not necessarily equal!
Equal vs. Equivalent Sets Comparison
Feature
Equal Sets ($A = B$)
Equivalent Sets ($A \approx B$)
Criterion
Identical elements
Identical count of elements ($n(A) = n(B)$)
Example
$\{1, 2, 3\} = \{3, 1, 2\}$
$\{1, 2, 3\} \approx \{a, b, c\}$
Bijection
Trivially identical mapping
Supports a one-to-one correspondence
Worked Benchmark Examples
Example 1 (Identifying Empty Sets):
State whether each of the following sets is Empty, Singleton, or neither:
(a) $A = \{x : x \in \mathbb{N}, 2 < x < 3\}$
(b) $B = \{x : x \in \mathbb{Z}, x^2 = 0\}$
(c) $C = \{x : x \text{ is an odd integer divisible by 2}\}$
(d) $D = \{\emptyset\}$
Show Step-by-Step Solution
• (a) $A$: Empty Set ($\emptyset$): There is no natural number strictly between 2 and 3.
• (b) $B$: Singleton Set ($\{0\}$): Only $x = 0$ satisfies $x^2 = 0$, and 0 is an integer. Cardinality is 1.
• (c) $C$: Empty Set ($\emptyset$): By definition, odd numbers leave a remainder of 1 when divided by 2; none are divisible by 2.
• (d) $D$: Singleton Set: $D$ contains exactly one element, which happens to be the symbol $\emptyset$. Its cardinality is $n(D) = 1$.
Example 2 (Equal vs. Equivalent Verification):
Given:
$A = \{x : x \text{ is a letter in the word 'WOLF'}\}$
$B = \{x : x \text{ is a letter in the word 'FLOW'}\}$
$C = \{1, 2, 3, 4\}$
State whether pairs $(A, B)$ and $(A, C)$ are Equal or Equivalent.
Show Step-by-Step Solution
• $A = \{W, O, L, F\}$, $n(A) = 4$.
• $B = \{F, L, O, W\}$, $n(B) = 4$.
• $C = \{1, 2, 3, 4\}$, $n(C) = 4$.
• Pair $(A, B)$: Equal ($A = B$) and Equivalent: Both sets have the exact same 4 letters.
• Pair $(A, C)$: Equivalent ($A \approx C$) but NOT Equal: Both sets share the same cardinal count of 4 elements, but their elements are completely different (letters vs numbers).
CHECK YOUR PROGRESS 4.2
Graded textbook exercises on Types of Sets, Finiteness, and Equality from CBSE Class 11 Applied Mathematics.
Problem 1Null Set Identification
State whether the set $S = \{x : x \in \mathbb{R}, x^2 + 1 = 0\}$ is an empty set or not, giving justification.
Show Step-by-Step Solution
• $x^2 + 1 = 0 \implies x^2 = -1 \implies x = \pm \sqrt{-1} = \pm i$ (imaginary numbers).
• Since there is no real number whose square is negative, no $x \in \mathbb{R}$ satisfies the equation.
• Therefore, $S$ is an Empty Set ($∅$).
Problem 2Finite vs Infinite
Classify the following sets as Finite or Infinite:
(a) The set of lines parallel to the $X$-axis.
(b) The set of concentric circles passing through the origin.
(c) The set of prime numbers less than 99.
Show Step-by-Step Solution
• (a) Infinite: Infinitely many parallel lines can be drawn.
• (b) Infinite: Circles of continuously varying radii can pass through the origin.
• (c) Finite: There are exactly 25 prime numbers less than 99 ($n = 25$).
Problem 3Equality of Solution Sets
Are the sets $A = \{2, 3\}$ and $B = \{x : x^2 + 5x + 6 = 0\}$ equal? Justify.
Show Step-by-Step Solution
• Solve $x^2 + 5x + 6 = 0 \implies (x + 2)(x + 3) = 0 \implies x = -2, -3$.
• Thus $B = \{-2, -3\}$.
• While $A = \{2, 3\}$.
• Since $+2 \in A$ but $+2 \notin B$, the sets are NOT Equal ($A \neq B$). They are, however, Equivalent ($n(A) = n(B) = 2$).
Problem 4Singleton Trap
Is the set $A = \{x : x \in \mathbb{W}, x < 1\}$ an empty set? State its element and cardinality.
Show Step-by-Step Solution
• Whole numbers begin at 0: $\mathbb{W} = \{0, 1, 2, \dots\}$.
• The only whole number strictly less than 1 is 0.
• Thus $A = \{0\}$, which is a Singleton Set with cardinality $n(A) = 1$. It is NOT empty.
Problem 5Equivalence Criterion
If $A = \{x : x \text{ is a vowel in 'EDUCATION'}\}$ and $B = \{2, 4, 6, 8, 10\}$, determine whether $A$ and $B$ are equivalent.
Show Step-by-Step Solution
• Vowels in 'EDUCATION': E, U, A, I, O $\implies A = \{A, E, I, O, U\} \implies n(A) = 5$.
• $B = \{2, 4, 6, 8, 10\} \implies n(B) = 5$.
• Since $n(A) = n(B) = 5$, the sets $A$ and $B$ are Equivalent ($A \approx B$).
Comprehensive Practice Worksheet: Types of Sets
Graded problems matching CBSE Class 11 Applied Mathematics examination blueprint.
Q2: State whether the set of points on a circle of radius 5 cm is finite or infinite.
Show Answer & Scheme
Answer [1 Mark]:Infinite, because a continuous geometrical curve consists of uncountably infinite points.
SECTION B: 2-Mark Short Answer Problems
Q3: Differentiate between Equal sets and Equivalent sets with suitable examples.
Show Answer & Scheme
Differences [2 Marks]:
• Equal Sets: Must possess exactly the same elements (e.g. $\{1, 2\} = \{2, 1\}$).
• Equivalent Sets: Must possess the same number of elements ($n(A) = n(B)$), but the elements themselves can be completely distinct (e.g. $\{1, 2\} \approx \{x, y\}$).
Q4: Show that the set $A = \{x : x \in \mathbb{Z}, x^2 - 4 = 0\}$ and $B = \{-2, 2\}$ are equal sets.
Show Answer & Scheme
• $x^2 - 4 = 0 \implies x^2 = 4 \implies x = 2, -2$ [1 Mark].
• Since both $2$ and $-2$ are integers, $A = \{-2, 2\}$.
• Since $A$ and $B$ have identical elements, $\mathbf{A = B}$ [1 Mark].
SECTION C: 4-Mark Comprehensive Scenario Drill
Q5: A telecommunications network categorizes subscriber account groups:
• Group Alpha: $\{x : x \text{ is a prime factor of } 30\}$
• Group Beta: $\{x : x \in \mathbb{N}, x^3 - 10x^2 + 31x - 30 = 0\}$
• Group Gamma: $\{x : x \in \mathbb{W}, x < 3\}$
(i) Find the elements of Group Alpha in Roster form.
(ii) Find the elements of Group Beta in Roster form.
(iii) Are Group Alpha and Group Beta equal sets?
(iv) Are Group Alpha and Group Gamma equal or equivalent sets?
Show Step-by-Step Solution & Marking Scheme
Part (i) Group Alpha [1 Mark]:
• Prime factors of 30 ($30 = 2 \times 3 \times 5$): $\mathbf{\{2, 3, 5\}}$.
Part (ii) Group Beta [1 Mark]:
• Test roots of $x^3 - 10x^2 + 31x - 30 = 0$:
$(x-2)(x-3)(x-5) = 0 \implies x = 2, 3, 5$: $\mathbf{\{2, 3, 5\}}$.
Part (iii) Alpha vs Beta [1 Mark]:
• Since both have identical elements $\{2, 3, 5\}$, Group Alpha = Group Beta (Equal Sets).
Part (iv) Alpha vs Gamma [1 Mark]:
• Group Gamma: Whole numbers $< 3$: $\{0, 1, 2\}$.
• $n(\text{Alpha}) = 3$ and $n(\text{Gamma}) = 3$.
• They are Equivalent sets ($n=3$), but NOT equal because $0, 1 \notin \text{Alpha}$.
High-Yield Quick Revision: Types of Sets
Summary of definitions, cardinality rules, and common boundary conditions for Chapter 4.2.
Set Classification Cheat Sheet
Set Type
Defining Property
Cardinality $n(A)$
Standard Example
Empty Set ($\emptyset$)
No elements exist
$0$
$\{x \in \mathbb{R} : x^2 + 4 = 0\}$
Singleton Set
Exactly one element
$1$
$\{x \in \mathbb{N} : x^2 = 9\} = \{3\}$
Finite Set
Countable terminal elements
$k \in \mathbb{W}$
$\{x \in \mathbb{N} : x < 100\}$
Infinite Set
Uncountable / non-terminating
$\infty$
$\mathbb{N}, \mathbb{Z}, \mathbb{R}$
Examination Alert: Empty vs. Singleton
Do not confuse $\emptyset$ with $\{\emptyset\}$ or $\{0\}$:
• $\emptyset$ has 0 elements (Empty Set).
• $\{\emptyset\}$ has 1 element (the symbol $\emptyset$ itself).
• $\{0\}$ has 1 element (the number 0).
10-Question Diagnostic Board Mock Examination: Types of Sets
Standard timed test: 4 Objective MCQs / Assertion-Reason with instant automatic grading, plus 6 multi-mark subjective problems with CBSE marking rubrics.
PART 1: Objective Section (4 Marks)
Q1: Which of the following is an empty set?
Solution: (C) $2x + 1 = 4 \implies 2x = 3 \implies x = 1.5$. Since $1.5 \notin \mathbb{N}$, this set is empty.
Q2: Two sets $A$ and $B$ are said to be equivalent if:
Solution: (B) Equivalent sets are defined by having the exact same cardinal number of elements: $n(A) = n(B)$.
Q3: The set of all circles passing through three non-collinear points is a:
Solution: (C) By basic Euclidean geometry, there is one and only one unique circle passing through three non-collinear points (Singleton set).
Q4 (Assertion-Reason): Assertion (A): All equal sets are equivalent, but all equivalent sets are not necessarily equal. Reason (R): Equal sets require identical elements, which inherently guarantees identical cardinality, whereas equivalent sets only require equal cardinality.
Solution: (A) Both Assertion and Reason are true and Reason provides the precise set theory relationship.
PART 2: Subjective Section (Q5 - Q10)
Q52 Marks
Examine whether the set $A = \{x : x \in \mathbb{N}, x^2 = 9\}$ and $B = \{x : x \in \mathbb{Z}, x^2 = 9\}$ are equal.
Show Step-by-Step Solution & Marking Scheme
• In $\mathbb{N}$: $x^2 = 9 \implies x = 3 \implies A = \{3\}$ [1 Mark].
• In $\mathbb{Z}$: $x^2 = 9 \implies x = 3, -3 \implies B = \{-3, 3\}$ [1 Mark].
• Since $-3 \in B$ but $-3 \notin A$, $\mathbf{A \neq B}$ (Not equal).
Q62 Marks
State whether the set of points on the line segment $AB$ of length 5 cm is finite or infinite. Justify.
Show Step-by-Step Solution & Marking Scheme
Answer [2 Marks]:
It is an Infinite Set. A geometric line segment consists of a continuum of real numbers representing lengths, between any two of which there are infinitely many more real points.
Q73 Marks
Classify each of the following sets as empty, singleton, finite, or infinite:
(a) $\{x : x \in \mathbb{N}, x \text{ is even and prime}\}$
(b) $\{x : x \in \mathbb{Z}, 2x = 3\}$
(c) $\{x : x \in \mathbb{R}, 0 < x < 1\}$.
Show Step-by-Step Solution & Marking Scheme
• (a) $\{2\} \implies$ Singleton set ($n = 1$) [1 Mark].
• (b) $x = 1.5 \notin \mathbb{Z} \implies$ Empty set ($\emptyset$) [1 Mark].
• (c) Real numbers strictly between 0 and 1 form a continuum $\implies$ Infinite set [1 Mark].
Q83 Marks
Explain why the set $\{\emptyset\}$ has cardinality 1, while $\emptyset$ has cardinality 0.
Show Step-by-Step Solution & Marking Scheme
• $\emptyset$ is the empty set, defined as having no members: $n(\emptyset) = 0$ [1.5 Marks].
• $\{\emptyset\}$ is a set that contains one element (namely, the empty set as a member entity). Its cardinality is $n(\{\emptyset\}) = 1$ [1.5 Marks].
• Analogy: An empty box ($emptyset$) vs. a box containing an empty box (${emptyset}$).
Q9 & Q10Case Study (4 Marks Total)
Case Context: Software Permissions Access Groups
A cloud security architect configures user role groups:
• Group $P_1 = \{x : x \text{ is a root admin with ID } x \in \mathbb{N}, x^2 - 16 = 0\}$
• Group $P_2 = \{x : x \text{ is a guest account with ID } x \in \mathbb{N}, x < 1\}$
Q9 (2 Marks): Identify the type of set represented by Group $P_1$ and state its elements. Q10 (2 Marks): Identify the type of set represented by Group $P_2$ and justify why it contains no active accounts.
Show Step-by-Step Solution & Marking Scheme
Solution to Q9 [2 Marks]:
• $x^2 - 16 = 0 \implies x = \pm 4$. Since $x \in \mathbb{N}$, only $x = 4$ is allowed.
• $P_1 = \{4\}$, which is a Singleton Set with cardinality 1.
Solution to Q10 [2 Marks]:
• Natural numbers begin at 1 ($1, 2, 3, \dots$). There is no natural number $< 1$.
• Therefore, $P_2 = \emptyset$, which is an Empty Set containing 0 accounts.