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CDS Sets, Relations & Functions

Study Material — 1 PYQs (2019–2019) · Concept Notes · Shortcuts

CDS Sets, Relations & Functions is a frequently tested subtopic — 1 previous year questions from 2019–2019 papers are included below with concept notes, key rules and shortcut tricks.

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2019–2019
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Previous Year Questions

CDS Sets, Relations & Functions — Past Exam Questions

1 questions from actual CDS papers · all shown free · click option to reveal solution

Exam Q 12019Previous Year Pattern

Let A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8}. A relation R is defined from A to B such that R = {(a, b) : a ∈ A, b ∈ B, and a divides b}. Which of the following statements is correct?

Concept Notes

Sets, Relations & Functions— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic

Sets, Relations and Functions form the foundation of modern mathematics and are crucial for NDA exam success. This topic appears in 2-3 questions every year, making it a high-scoring area when mastered properly. CORE CONCEPT

A Set is a collection of distinct objects. A Relation connects elements from one set to another. A Function is a special relation where each input has exactly one output. Think of a function as a machine - you put something in, you get exactly one thing out. **

2.Key Rules

Core rules you must know cold

Set Operations: Union (∪), Intersection (∩), Complement ('), Difference (-) For any sets A and B: n(A∪B) = n(A) + n(B) - n(A∩B) Function Types: One-to-One (Injective), Onto (Surjective), Bijective (both) Domain: Input values, Range: Output values, Codomain: Possible output set

3.Formula Block

Memorise — at least one formula appears in every paper
n(A∪B∪C) = n(A) + n(B) + n(C) - n(A∩B) - n(B∩C) - n(A∩C) + n(A∩B∩C)
For complement: n(A') = n(U) - n(A) where U is universal set
Cartesian Product: n(A×B) = n(A) × n(B)
Number of relations from A to B = 2^(n(A)×n(B))
Number of functions from A to B = [n(B)]^n(A)

4.Exam Patterns

What examiners ask — read before attempting PYQs

NDA typically asks: Venn diagram problems (40% questions), function domain/range identification (30% questions), relation properties verification (20% questions), and set operation calculations (10% questions). Most questions are direct application based, not proof-based.

5.Shortcuts

Use these to save 30–60 seconds per question

** For three sets A, B, C: Draw overlapping circles. Start filling from center (A∩B∩C), then work outward. Use: Only A = n(A) - n(A∩B) - n(A∩C) + n(A∩B∩C).

This eliminates calculation errors. **

6.Worked Example

Solve this step-by-step before moving on
1
Step 1

Given n(U) = 50, n(C) = 30, n(F) = 25, n(C∩F) = 10

2
Step 2

Apply formula n(C∪F) = n(C) + n(F) - n(C∩F)

3
Step 3

n(C∪F) = 30 + 25 - 10 = 45

4
Step 4

Students playing neither = n(U) - n(C∪F) = 50 - 45 = 5 Answer: 5 students play neither game. WORKED EXAMPLE 2 Find domain of f(x) = √(x-2)/(x-3)

1
Step 1

For square root, expression inside must be ≥ 0, so x-2 ≥ 0, giving x ≥ 2

2
Step 2

For fraction, denominator cannot be zero, so x-3 ≠ 0, giving x ≠ 3

3
Step 3

Combine conditions: x ≥ 2 and x ≠ 3

4
Step 4

Domain = [2,3) ∪ (3,∞) Answer: All real numbers greater than or equal to 2, except 3. **

7.Quick Trick

Fast identification technique

Quick test for function: Draw vertical lines on graph. If any vertical line cuts the curve more than once, it's NOT a function. For one-to-one: use horizontal line test.

8.Exam Traps

Common mistakes students make — avoid these

** Students confuse 'into' and 'onto' functions. Remember: A function is ONTO when every element in codomain has a pre-image. It's INTO when some codomain elements have no pre-image.

Always check if range equals codomain for onto functions. RELATION PROPERTIES SHORTCUT Reflexive: (a,a) exists for all a Symmetric: If (a,b) exists, then (b,a) exists Transitive: If (a,b) and (b,c) exist, then (a,c) exists Mnemonic: 'RST' - Reflexive Symmetric Transitive makes equivalence relation.

Key Points to Remember

  • Set union formula: n(A∪B) = n(A) + n(B) - n(A∩B)
  • A function has exactly one output for each input value
  • Domain is input set, range is actual output set, codomain is possible output set
  • Number of subsets of a set with n elements = 2^n
  • Cartesian product n(A×B) = n(A) × n(B)
  • Onto function: range = codomain, Into function: range ⊂ codomain
  • Equivalence relation must be reflexive, symmetric and transitive
  • Number of functions from set A to set B = [n(B)]^n(A)
  • Complement formula: n(A') = n(U) - n(A)
  • For function graphs, vertical line test determines if relation is function

Exam-Specific Tips

  • Empty set ∅ is subset of every set
  • Number of relations from set A to set B = 2^(n(A)×n(B))
  • Identity function f(x) = x has domain = range = real numbers
  • Modulus function f(x) = |x| has domain = R, range = [0,∞)
  • Greatest integer function [x] has domain = R, range = integers
  • A set with n elements has 2^n subsets and 2^n - 1 proper subsets
  • Universal set U contains all elements under consideration in given context
  • Bijective function is both one-to-one and onto simultaneously
Practice MCQs

Sets, Relations & Functions — Practice Questions

39graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 39

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Practice 1easy

Let A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8}. If R is a relation from A to B defined by R = {(a, b) : a ∈ A, b ∈ B, and b = 2a}, then the number of elements in R is:

Practice 2easy

Let f: A → B and g: B → C be two functions. If g ∘ f is injective, which of the following must be true?

Practice 3easy

Let A = {x ∈ ℤ : −3 ≤ x ≤ 3} and B = {x ∈ ℤ : x² < 9}. Then A ∩ B is equal to:

Practice 4easy

Let f: {1, 2, 3, 4} → {a, b, c, d} be a function defined by f(1) = a, f(2) = b, f(3) = c, f(4) = d. Which of the following statements is correct?

Practice 5easy

Let R be a relation on the set A = {1, 2, 3, 4} defined by R = {(a, b) : a − b is even}. Which of the following is true about R?

Practice 6easy

Let f: ℝ → ℝ be defined by f(x) = 2x + 3. If g: ℝ → ℝ is defined by g(x) = x² − 1, then (f ∘ g)(2) is equal to:

Practice 7easy

Let A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8}. If R is a relation from A to B defined by R = {(a, b) : a ∈ A, b ∈ B, and a divides b}, then the number of elements in R is:

Practice 8easy

Let f: ℝ → ℝ be defined by f(x) = x² − 4x + 5. Which of the following statements is correct?

Practice 9easy

Let A = {1, 2, 3} and B = {a, b}. The number of relations from A to B is:

Practice 10easy

Let f: {1, 2, 3, 4} → {a, b, c} be a function. If f(1) = a, f(2) = b, f(3) = c, and f(4) = a, then which of the following is true?

Practice 11easy

Let R be a relation on the set A = {1, 2, 3, 4} defined by R = {(a, b) : a − b is even}. Which of the following ordered pairs is NOT in R?

Practice 12easy

Let f: ℝ → ℝ be defined by f(x) = x² + 2x + 3. Which of the following statements is correct?

Practice 13easy

Let A and B be two finite sets such that n(A) = 5, n(B) = 3, and n(A ∩ B) = 2. Then n(A ∪ B) is equal to:

Practice 14easy

Let R be a relation on the set ℕ (natural numbers) defined by R = {(a, b) : a, b ∈ ℕ and a divides b}. Which of the following properties does R satisfy?

Practice 15medium

Let f: ℝ → ℝ be defined by f(x) = (x² - 4)/(x - 2) for x ≠ 2. The domain of f is ℝ \ {2}. If we define g(x) = x + 2 for all x ∈ ℝ, then which of the following statements is correct?

Practice 16medium

Let f: ℝ → ℝ be defined by f(x) = (x² - 4)/(x - 2) for x ≠ 2. If we extend f to a function g: ℝ → ℝ by defining g(2) = L, then g is continuous at x = 2. The value of L is:

Practice 17medium

Let R be a relation on the set A = {1, 2, 3, 4} defined by R = {(a, b) : |a - b| is even}. Which of the following statements is correct?

Practice 18medium

Let A = {x ∈ ℝ : x² - 5x + 6 ≤ 0} and B = {x ∈ ℝ : |x - 2| < 2}. Then A ∩ B is equal to:

Practice 19medium

Let f: ℝ → ℝ be defined by f(x) = (2x + 3)/(x - 1). The range of f is:

Practice 20medium

Let A = {x ∈ ℝ : x² - 5x + 6 ≤ 0} and B = {x ∈ ℝ : |x - 2| < 2}. Then A ∩ B is:

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60-Second Revision — Sets, Relations & Functions

  • Remember: n(A∪B) = n(A) + n(B) - n(A∩B) for all Venn diagram problems
  • Formula: Number of functions from A to B = [n(B)]^n(A)
  • Trap: Don't confuse 'onto' (range = codomain) with 'into' (range ⊂ codomain)
  • Quick check: Vertical line test for functions, horizontal line test for one-to-one
  • Remember: Equivalence relation needs reflexive + symmetric + transitive properties
  • Formula: Set with n elements has 2^n total subsets
  • Key point: Domain restrictions come from denominators ≠ 0 and square roots ≥ 0
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