Comprehensive guide and practice questions for CBSE Class 11 Applied Mathematics curriculum (2026-27).
Comprehensive Concepts & Principles: Sets and Representation
In mathematics and quantitative analytics, a Set is a well-defined collection of distinct objects, termed elements or members. The phrase well-defined means that given any object, there is an objective, unequivocal criterion to decide whether it belongs to the collection or not. In CBSE Class 11 Applied Mathematics, set theory provides the foundational language for database querying, probability sample spaces, categorical market segmentation, and relational models.
Key Set Theory Foundations:
Membership Notation: If an element $x$ belongs to set $A$, we write $x \in A$. If $x$ does not belong to set $A$, we write $x \notin A$.
Well-Defined vs. Subjective Collections: • Well-Defined (A Set): "The collection of all natural numbers less than 10" or "All vowels in the English alphabet".
• Subjective (Not a Set): "The collection of the 5 most talented batsmen in the world" (talent is subjective without strict statistical criteria).
Standard Mathematical Sets: • $\mathbb{N}$: The set of all natural numbers $\{1, 2, 3, \dots\}$
• $\mathbb{W}$: The set of all whole numbers $\{0, 1, 2, 3, \dots\}$
• $\mathbb{Z}$: The set of all integers $\{\dots, -2, -1, 0, 1, 2, \dots\}$
• $\mathbb{Q}$: The set of all rational numbers $\{\frac{p}{q} : p, q \in \mathbb{Z}, q \neq 0\}$
• $\mathbb{R}$: The set of all real numbers
Methods of Representing Sets
A set can be mathematically specified in two primary formats:
1. Roster (Tabular) Form
All elements are listed explicitly inside curly braces $\{ \}$, separated by commas:
• $V = \{a, e, i, o, u\}$
• $P = \{2, 3, 5, 7, 11\}$ Rules: Order of listing is immaterial ($\{1, 2\} = \{2, 1\}$); elements are never duplicated ($\{\text{M, A, T, H, E, I, C, S}\}$ for 'MATHEMATICS').
2. Set-Builder Form
Elements are described by characterizing a common defining rule/property $P(x)$:
• $A = \{x : x \in \mathbb{N} \text{ and } x < 6\}$
• $B = \{x : x = 2n, n \in \mathbb{N}, n \le 5\}$ Read as: "The set of all $x$ such that $x$ satisfies property $P$."
Worked Benchmark Examples
Example 1 (Converting Set-Builder to Roster Form):
Write each of the following sets in Roster (Tabular) form:
(a) $A = \{x : x \text{ is an integer and } -3 \le x < 5\}$
(b) $B = \{x : x \in \mathbb{N} \text{ and } x^2 - 5x + 6 = 0\}$
(c) $C = \{x : x \text{ is a two-digit natural number such that the sum of its digits is 8}\}$
Show Step-by-Step Solution
• (a) $A$: Integers from $-3$ up to (but excluding) $5$:
$$\mathbf{A = \{-3, -2, -1, 0, 1, 2, 3, 4\}}$$
• (b) $B$: Solve $x^2 - 5x + 6 = 0 \implies (x - 2)(x - 3) = 0 \implies x = 2, 3$. Since both are natural numbers:
$$\mathbf{B = \{2, 3\}}$$
• (c) $C$: Two-digit numbers where digits add to 8: $17, 26, 35, 44, 53, 62, 71, 80$:
$$\mathbf{C = \{17, 26, 35, 44, 53, 62, 71, 80\}}$$
Example 2 (Converting Roster to Set-Builder Form):
Express the following sets in Set-Builder form:
(a) $S = \{1, 4, 9, 16, 25, \dots, 100\}$
(b) $T = \left\{\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}\right\}$
Show Step-by-Step Solution
• (a) $S$: Elements are perfect squares of natural numbers from $1^2$ to $10^2$:
$$\mathbf{S = \{x : x = n^2, n \in \mathbb{N}, 1 \le n \le 10\}}$$
• (b) $T$: Each fraction has numerator $n$ and denominator $n+1$ for $n = 1, 2, \dots, 6$:
$$\mathbf{T = \left\{x : x = \frac{n}{n+1}, n \in \mathbb{N}, 1 \le n \le 6\right\}}$$
CHECK YOUR PROGRESS 4.1
Graded textbook exercises on Set Concepts, Well-Defined Criteria, and Dual Representations from CBSE Class 11 Applied Mathematics.
Problem 1Well-Defined Evaluation
Which of the following collections represent valid mathematical sets? Give reasons:
(a) The collection of all prime numbers less than 20.
(b) The collection of all difficult chapters in Class 11 Applied Mathematics.
(c) The collection of all months of a year beginning with the letter 'J'.
Show Step-by-Step Solution
• (a) Is a Set: Whether a number is prime and $< 20$ is mathematically definitive: $\{2, 3, 5, 7, 11, 13, 17, 19\}$.
• (b) Not a Set: "Difficult" is a subjective judgment that varies from student to student without an objective boundary.
• (c) Is a Set: Well-defined: $\{\text{January, June, July}\}$.
Problem 2Roster Conversion
Write the set of all letters in the word "COMMISSION" in Roster form.
Show Step-by-Step Solution
• In roster form, duplicated letters are written only once.
• Unique letters: C, O, M, I, S, N.
• $\mathbf{\{C, O, M, I, S, N\}}$.
Problem 3Set-Builder Pattern
Write the set $D = \{5, 25, 125, 625\}$ in Set-Builder form.
Show Step-by-Step Solution
• Elements are powers of 5: $5^1, 5^2, 5^3, 5^4$.
• $\mathbf{D = \{x : x = 5^n, n \in \mathbb{N}, 1 \le n \le 4\}}$.
Problem 4Equation Solution Set
Write the solution set of the equation $x^2 + x - 2 = 0$ in roster form for $x \in \mathbb{R}$.
Show Step-by-Step Solution
• $x^2 + x - 2 = 0 \implies (x + 2)(x - 1) = 0 \implies x = 1, -2$.
• Both roots are real numbers.
• $\mathbf{\{-2, 1\}}$.
Problem 5Membership Notation
Let $A = \{1, 2, \{3, 4\}, 5\}$. State whether each statement is True or False:
(a) $\{3, 4\} \in A$
(b) $3 \in A$
(c) $5 \in A$.
Show Step-by-Step Solution
• (a) TRUE: The set $\{3, 4\}$ is itself an element of $A$.
• (b) FALSE: The individual number 3 is an element of $\{3, 4\}$, but not a direct member of $A$.
• (c) TRUE: 5 is explicitly listed as a direct member of $A$.
Comprehensive Practice Worksheet: Sets & Representation
Graded problems matching CBSE Class 11 Applied Mathematics examination blueprint.
Definition [1 Mark]: A set is a well-defined collection of distinct objects.
Q2: Represent the set $E = \{x : x \text{ is an even prime number}\}$ in roster form.
Show Answer & Scheme
$$\mathbf{\{2\}} \quad [1 \text{ Mark}]$$
SECTION B: 2-Mark Short Answer Problems
Q3: State the two primary rules to be followed when writing a set in Roster (Tabular) form.
Show Answer & Scheme
Rules [2 Marks]:
(1) No Duplication: An element is not repeated; each distinct member is listed once.
(2) Order Immaterial: The sequence or order in which elements are listed inside the braces does not alter the set.
Q4: Write the set $A = \{2, 4, 8, 16, 32\}$ in set-builder form.
Show Answer & Scheme
$$\mathbf{A = \{x : x = 2^n, n \in \mathbb{N}, 1 \le n \le 5\}} \quad [2 \text{ Marks}]$$
SECTION C: 4-Mark Comprehensive Scenario Drill
Q5: An e-commerce platform stores customer loyalty tiers based on annual order counts ($n \in \mathbb{N}$):
• Tier Alpha: $\{n : n \in \mathbb{N}, n \le 5\}$
• Tier Beta: $\{n : n = 2k + 1, k \in \mathbb{W}, 3 \le k \le 7\}$
• Tier Gamma: $\{n : n \in \mathbb{N}, n^2 - 12n + 35 = 0\}$
(i) Express Tier Alpha in Roster form.
(ii) Express Tier Beta in Roster form.
(iii) Express Tier Gamma in Roster form.
(iv) Identify if any order count qualifies a customer for both Tier Beta and Tier Gamma simultaneously.
Show Step-by-Step Solution & Marking Scheme
Part (i) Tier Alpha [1 Mark]: Natural numbers $\le 5$: $\mathbf{\{1, 2, 3, 4, 5\}}$. Part (ii) Tier Beta [1 Mark]: $k = 3, 4, 5, 6, 7 \implies n = 7, 9, 11, 13, 15$: $\mathbf{\{7, 9, 11, 13, 15\}}$. Part (iii) Tier Gamma [1 Mark]: $n^2 - 12n + 35 = 0 \implies (n-5)(n-7) = 0 \implies n = 5, 7$: $\mathbf{\{5, 7\}}$. Part (iv) Overlap [1 Mark]: The value $\mathbf{n = 7}$ belongs to both Tier Beta and Tier Gamma.
High-Yield Quick Revision: Sets & Representation
Summary of notation, representation modes, and standard number systems for Chapter 4.1.
Roster vs. Set-Builder Comparison
Feature
Roster Form
Set-Builder Form
Format
$\{a_1, a_2, a_3, \dots\}$
$\{x : P(x)\}$
Best Used For
Small finite sets or clear enumerations
Large/infinite sets governed by a mathematical formula
Example
$\{2, 4, 6, 8\}$
$\{x : x = 2n, n \in \mathbb{N}, n \le 4\}$
Critical Examination Trap
Never duplicate elements when writing words in roster form! For instance, the letters of "BETTER" must be written as $\{B, E, T, R\}$, not $\{B, E, T, T, E, R\}$.
Timed exam: 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 a well-defined set?
Solution: (C) Prime numbers less than 15 have an objective, unambiguous mathematical definition.
Q2: The set $\{x : x \in \mathbb{Z}, x^2 < 10\}$ in roster form is:
Solution: (C) Integers whose squares are $< 10$ are $-3, -2, -1, 0, 1, 2, 3$.
Q3: The set of all letters in the word "TRIGONOMETRY" in tabular form has how many distinct elements?
Solution: (B) Unique letters: T, R, I, G, O, N, M, E, Y (9 distinct letters).
Q4 (Assertion-Reason): Assertion (A): The collection $\{1, 2, 2, 3\}$ and $\{1, 2, 3\}$ represent the exact same mathematical set. Reason (R): In set theory, the repetition of an element in roster form is redundant and does not change the membership of the set.
Solution: (A) Both Assertion and Reason are true and Reason provides the fundamental rule of set definition.
PART 2: Subjective Section (Q5 - Q10)
Q52 Marks
Write the set $A = \{x : x \in \mathbb{Z}, x^2 - 9 = 0\}$ in roster form.
Show Step-by-Step Solution & Marking Scheme
• $x^2 - 9 = 0 \implies x^2 = 9 \implies x = \pm 3$ [1 Mark].
• Both $+3$ and $-3$ are integers.
• $\mathbf{A = \{-3, 3\}}$ [1 Mark].
Q62 Marks
Write the set $B = \{1, 8, 27, 64, 125\}$ in set-builder form.
Show Step-by-Step Solution & Marking Scheme
• Elements are cubes of the first 5 natural numbers: $1^3, 2^3, 3^3, 4^3, 5^3$ [1 Mark].
• $\mathbf{B = \{x : x = n^3, n \in \mathbb{N}, 1 \le n \le 5\}}$ [1 Mark].
Q73 Marks
State whether the following are sets. Justify your answer mathematically:
(a) The collection of all honest politicians in India.
(b) The collection of all roots of the equation $2x^3 - 3x^2 - 11x + 6 = 0$.
(c) The collection of all intelligent students in a school.
Show Step-by-Step Solution & Marking Scheme
• (a) Not a set [1 Mark]: "Honest" is an uncalibrated subjective virtue, lacking an objective boundary.
• (b) Is a set [1 Mark]: The roots of a polynomial equation are mathematically determined and well-defined.
• (c) Not a set [1 Mark]: "Intelligent" is a qualitative relative adjective without standard defining criteria.
Q83 Marks
Let $A = \{x : x \text{ is a letter in the word 'MATHEMATICS'}\}$.
(i) Write $A$ in roster form.
(ii) Find the cardinal number $n(A)$.
(iii) State whether $P \in A$ and $M \in A$.
Show Step-by-Step Solution & Marking Scheme
• (i) Roster Form: $\mathbf{\{M, A, T, H, E, I, C, S\}}$ [1 Mark].
• (ii) Cardinal Number: Count of distinct elements is $\mathbf{n(A) = 8}$ [1 Mark].
• (iii) $P \notin A$ (False) and $M \in A$ (True) [1 Mark].
Q9 & Q10Case Study (4 Marks Total)
Case Context: Relational Database Query Filtering
A relational database engineer defines three SQL customer record sets from table `Orders` with integer Order IDs ($x \in \mathbb{N}$):
• $S_1 = \{x : x = 3k, k \in \mathbb{N}, k \le 6\}$
• $S_2 = \{x : x = 2m, m \in \mathbb{N}, m \le 9\}$
Q9 (2 Marks): List both sets $S_1$ and $S_2$ in Roster form. Q10 (2 Marks): Write the set of Order IDs that belong to BOTH $S_1$ and $S_2$ in set-builder form.
Solution to Q10 [2 Marks]:
• Common elements are multiples of both 2 and 3 (i.e. multiples of 6) up to 18: $\{6, 12, 18\}$.
• In Set-Builder form:
$$\mathbf{\{x : x = 6n, n \in \mathbb{N}, 1 \le n \le 3\}}$$