Chapter 1 Relations Class 12

Class 12 Mathematics Study Material
Chapter 1: Relations

Author: CBSE NCERT Curriculum

1. Ordered Pairs and Cartesian Product of Sets

Definition: An ordered pair is a pair of elements (a, b) where the order matters. (a, b) ≠ (b, a) unless a = b.

  • Used in coordinate geometry: (x, y) represents a point.
  • In real life: (Latitude, Longitude) represents a location.

Definition: The Cartesian product of two sets A and B, written as A × B, is the set of all ordered pairs (a, b) such that a ∈ A and b ∈ B.

Important Points:
  • If A has m elements and B has n elements, then A × B has m × n elements.
  • A × (B ∪ C) = (A × B) ∪ (A × C)
  • A × (B ∩ C) = (A × B) ∩ (A × C)
  • n(A × B) = n(A) × n(B)
Solved Example: If A = {1, 2} and B = {3, 4}, find A × B.
A × B = {(1,3), (1,4), (2,3), (2,4)}
Solved Example: If X = {a, b} and Y = {1, 2, 3}, find Y × X.
Y × X = {(1,a), (1,b), (2,a), (2,b), (3,a), (3,b)}
Solved Example: If A = {1, 2, 3} and B = {3, 4}, find A × B and B × A.
A × B = {(1,3), (1,4), (2,3), (2,4), (3,3), (3,4)}
B × A = {(3,1), (3,2), (3,3), (4,1), (4,2), (4,3)}
Solved Example: If A × B = {(1,2), (1,3), (2,2), (2,3)}, find sets A and B.
A = {1,2}, B = {2,3}
Practice Problems:
  1. If A = {x, y}, B = {1, 2, 3}, list A × B.
    Answer: {(x,1), (x,2), (x,3), (y,1), (y,2), (y,3)}
  2. If A has 4 elements and B has 3, how many elements in A × B?
    Answer: 12
  3. True or False: (2, 5) = (5, 2)
    Answer: False
  4. A = {x, y, z}, B = {1, 2}, find A × B and B × A.
    A × B = {(x,1), (x,2), (y,1), (y,2), (z,1), (z,2)}
    B × A = {(1,x), (1,y), (1,z), (2,x), (2,y), (2,z)}
  5. If A × B = {(a,1), (a,2), (b,1), (b,2)}, find A and B.
    Answer: A = {a,b}, B = {1,2}
  6. Prove A × (B ∩ C) = (A × B) ∩ (A × C)

2. Concept of Relation

Definition: A relation R from set A to B is a subset of the Cartesian product A × B.

  • If (a, b) ∈ R, we write aRb.
  • Relations model real-world ideas: “is father of”, “is greater than”, etc.
  • If R ⊆ A × B, then R is a relation from A to B.
Solved Example: Let A = {1, 2, 3}, B = {4, 5}, define R where x + y = 6.
R = {(1,5), (2,4)}
Solved Example: A = {1,2,3,4}, R = {(x,y) | y = x+1}.
R = {(1,2), (2,3), (3,4)}
Solved Example: A = {1,2,3,4,5}, R = {(x,y) | x = y²}.
R = {(1,1), (4,2)}
Solved Example: How many relations from A = {1,2,3} to B = {4,5}?
n(A × B) = 6 → 2⁶ = 64 relations
Practice Problems:
  1. A = {1,2,3}, B = {2,4,6}, define R = {(a,b) | b = 2a}.
    Answer: {(1,2), (2,4), (3,6)}
  2. Total relations from A (3 elements) to B (4 elements)?
    Answer: 2¹² = 4096
  3. If A has n elements, how many relations on A?
    Answer: 2ⁿ²
  4. Prove R = {(a,b) | a – b even} on ℤ is reflexive.
    Answer: a - a = 0, which is even
  5. Example of relation symmetric but not transitive:
    Answer: R = {(1,2), (2,1), (2,3), (3,2)}
Domain, Codomain and Range of a Relation
Definition:
  • Domain: Set of all first elements (a) such that (a, b) ∈ R
  • Codomain: The set B
  • Range: Set of all second elements (b) such that (a, b) ∈ R
Remember:
  • Range ⊆ Codomain, Domain ⊆ A
  • Domain is all first elements; Range is only those second elements that appear.
Solved Example:
Given R = {(1,2), (2,3), (3,3)}
Domain = {1,2,3}, Range = {2,3}
Find domain and range of R = {(1,4), (2,5), (3,6)}
Domain = {1,2,3}, Range = {4,5,6}
Practice:
  • Domain & Range of R = {(2,5), (3,6), (2,7)} → Domain = {2,3}, Range = {5,6,7}
  • Max pairs if domain = {1,2,3}, range = {4,5} → 3 × 2 = 6
  • R = {(x, y) | y = 2x}, x ∈ N → Domain = N, Range = {2, 4, 6, …}
Types of Relations
Reflexive Relation:
A relation R on A is reflexive if (a, a) ∈ R for every a ∈ A.
Example: R = {(1,1), (2,2), (3,3), (1,2)} is reflexive
Symmetric Relation:
R is symmetric if (a, b) ∈ R ⇒ (b, a) ∈ R.
Example: R = {(1,2), (2,1), (3,3)} is symmetric
Transitive Relation:
R is transitive if (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R.
Example: R = {(1,2), (2,3), (1,3)} is transitive
Equivalence Relation:
A relation that is reflexive, symmetric, and transitive.
Example: Congruence modulo n on ℤ
Important Formulas:
  • Number of reflexive relations: 2n² – n
  • Number of symmetric relations: 2n(n+1)/2
  • Equivalence Relation ⇒ Partitions the set

Solved Examples

Example:

Let A = {1, 2, 3} and R = {(1,1), (2,2), (3,3), (1,2), (2,1)}. Show if R is reflexive, symmetric, transitive.

  • Reflexive: Yes, all (a,a) are present.
  • Symmetric: (1,2) and (2,1) both present.
  • Transitive: (1,2) and (2,1) imply (1,1), which is present.

⇒ R is an equivalence relation.

Example:

Check whether the relation R in ℝ defined by R = {(a, b) : a ≤ b} is reflexive, symmetric, and transitive.

  • Reflexive: Yes, since a ≤ a for all a ∈ ℝ
  • Symmetric: No, because 2 ≤ 3 but 3 ≰ 2
  • Transitive: Yes, because a ≤ b and b ≤ c ⇒ a ≤ c

Practice Problems

  1. Define a reflexive but not symmetric relation on A = {1,2}.
    Answer: R = {(1,1), (2,2), (1,2)}
  2. Is the relation R = {(a,b) | a ≤ b} symmetric?
    Answer: No
  3. Find whether R = {(1,1), (2,2), (1,2)} is transitive.
    Answer: Yes
  4. Let A = {1, 2, 3}. Determine whether R = {(1,1), (2,2)} is reflexive.
    Answer: No (missing (3,3))
  5. Show that the relation R in ℤ defined by R = {(a, b): a – b is divisible by 5} is an equivalence relation.
    Answer: Reflexive (a-a = 0 divisible by 5), Symmetric (if a-b divisible by 5, then b-a also divisible by 5), Transitive (a-b and b-c ⇒ a-c divisible by 5)
  6. How many equivalence relations can be defined on a 3-element set?
    Answer: 5
  7. Prove that the relation “is parallel to” on the set of all lines in a plane is an equivalence relation.
    Answer: Reflexive (every line is parallel to itself), Symmetric (if L ∥ M then M ∥ L), Transitive (if L ∥ M and M ∥ N then L ∥ N)

Chapter-End MCQ Test: Relations

1. The number of elements in A × B when |A| = 4, |B| = 5 is:

  • (A) 9
  • (B) 20
  • (C) 10
  • (D) 1

2. If A = {x, y} and B = {1, 2}, then B × A is:

  • (A) {(x,1), (x,2), (y,1), (y,2)}
  • (B) {(1,x), (1,y), (2,x), (2,y)}
  • (C) {(x,x), (y,y)}
  • (D) {(1,2), (2,1)}

3. A relation R defined on a set A is said to be symmetric if:

  • (A) (a,a) ∈ R
  • (B) (a,b) ∈ R ⇒ (b,a) ∈ R
  • (C) (a,b),(b,c) ∈ R ⇒ (a,c) ∈ R
  • (D) None of these

4. If A = {1, 2, 3}, then the number of relations on A is:

  • (A) 2³
  • (B) 3²
  • (C) 2⁹
  • (D) 9²

5. The set of all ordered pairs from A = {1, 2} and B = {a} is:

  • (A) {(a,1), (a,2)}
  • (B) {(1,a), (2,a)}
  • (C) {(1,2), (a,a)}
  • (D) {(1,1), (2,2)}

6. The relation “is equal to” is:

  • (A) Reflexive
  • (B) Symmetric
  • (C) Transitive
  • (D) All of these

7. Number of reflexive relations on a set with 3 elements is:

  • (A) 8
  • (B) 27
  • (C) 64
  • (D) 512

8. A relation R on ℝ defined by aRb ⇔ a – b ∈ ℚ is:

  • (A) Not symmetric
  • (B) Not transitive
  • (C) Not reflexive
  • (D) Equivalence

9. The domain of R = {(1,2), (3,4), (5,6)} is:

  • (A) {1, 3, 5}
  • (B) {2, 4, 6}
  • (C) {1, 2, 3, 4, 5, 6}
  • (D) {6, 4, 2}

10. Let A = {1,2}, B = {a,b}. Total number of relations from A to B is:

  • (A) 4
  • (B) 8
  • (C) 16
  • (D) 32

11. A relation R is reflexive on A if:

  • (A) (a,a) ∉ R for all a ∈ A
  • (B) (a,b) ∈ R ⇒ (b,a) ∈ R
  • (C) (a,a) ∈ R for all a ∈ A
  • (D) None of these

12. A relation that is reflexive, symmetric and transitive is called:

  • (A) Partial Order
  • (B) Equivalence Relation
  • (C) Function
  • (D) Binary Relation

13. The number of symmetric relations on a set with n elements is:

  • (A) 2n(n+1)/2
  • (B) n²
  • (C) 2n
  • (D) 2n²

14. Let R = {(a,b) | a + b = 10}. Then (3,7) ∈ R:

  • (A) Yes
  • (B) No
  • (C) Sometimes
  • (D) Can’t say

15. If R is a relation on ℕ defined by aRb ⇔ a divides b, then R is:

  • (A) Reflexive and transitive
  • (B) Symmetric only
  • (C) Transitive only
  • (D) Not a relation

16. The relation R = {(1,1), (2,2), (1,2), (2,1)} on {1,2} is:

  • (A) Reflexive
  • (B) Symmetric
  • (C) Transitive
  • (D) All of these

17. A set of all relations from A = {1, 2} to B = {a} has how many elements?

  • (A) 2
  • (B) 4
  • (C) 8
  • (D) 16

18. If R = {(x, y) | x² + y² = 25}, then (3,4) ∈ R?

  • (A) Yes
  • (B) No
  • (C) Only if x > y
  • (D) Can’t say

19. Number of antisymmetric relations on a set with 2 elements:

  • (A) 5
  • (B) 6
  • (C) 9
  • (D) 10

20. If R = {(a,b) | a – b ∈ ℤ} on ℚ, then R is:

  • (A) Reflexive and symmetric
  • (B) Reflexive only
  • (C) Equivalence
  • (D) Symmetric only
Chapter-End MCQ Test: Relations

Self-Assessment Test

Chapter: Relations
Class 12 CBSE

General Instructions
  • All questions are compulsory.
  • This test consists of four sections: A, B, C and D.
  • Read the questions carefully and attempt as directed.

Section A: Multiple Choice Questions (1 mark each)

  1. If A = {1, 2, 3} and B = {a, b}, then the number of elements in A × B is:
    • (A) 3
    • (B) 2
    • (C) 6
    • (D) 9
  2. A relation R on a set A is said to be reflexive if:
    • (A) (a,b) ∈ R ⇒ (b,a) ∈ R
    • (B) (a,b),(b,c) ∈ R ⇒ (a,c) ∈ R
    • (C) (a,a) ∈ R for all a ∈ A
    • (D) (a,a) ∉ R for any a ∈ A
  3. The relation R = {(x,y) | x+y = 10} is defined on natural numbers. Then (3,7) ∈ R is:
    • (A) True
    • (B) False
    • (C) Not defined
    • (D) Depends on x
  4. The total number of relations on a set A with n elements is:
    • (A) 2ⁿ
    • (B) n²
    • (C) 2ⁿ²
    • (D) nⁿ
  5. The set of all ordered pairs from A = {1, 2} and B = {3} is:
    • (A) {(1,3), (2,3)}
    • (B) {(3,1), (3,2)}
    • (C) {(1,1), (2,2)}
    • (D) {(3,3), (1,2)}
  6. A relation R on ℕ defined by aRb if a divides b is:
    • (A) Symmetric
    • (B) Reflexive and transitive
    • (C) Anti-symmetric
    • (D) Equivalence
  7. The relation “is parallel to” among lines in a plane is:
    • (A) Symmetric only
    • (B) Reflexive and symmetric
    • (C) Reflexive, symmetric and transitive
    • (D) None of these
  8. If R is defined on ℝ by aRb if a – b ∈ ℚ, then R is:
    • (A) Not reflexive
    • (B) Not symmetric
    • (C) Not transitive
    • (D) Equivalence

Section B: Short Answer Questions (2 marks each)

  1. Let A = {1,2,3} and define relation R = {(1,1), (2,2), (3,3), (1,2), (2,1)}. Is R symmetric?
  2. Determine whether the relation R = {(a,b) ∈ ℤ × ℤ : a – b is even} is an equivalence relation.
  3. If A = {1,2} and B = {3,4}, write all elements of A × B and state its cardinality.
  4. Define a relation R on the set A = {1,2,3} as R = {(a,b): a ≤ b}. List all ordered pairs of R.
  5. Let A = {1,2} and define R on A by R = {(1,1), (2,2)}. Is R transitive?
  6. Show that the relation R = {(a,b) ∈ ℝ² : a – b = 0} is reflexive, symmetric and transitive.

Section C: Long Answer Questions (4 marks each)

  1. Let A = {1,2,3} and relation R is defined by R = {(1,1), (2,2), (3,3), (1,2), (2,1)}. Check whether the relation R is reflexive, symmetric, and transitive.
  2. Define a relation R on ℤ by aRb if and only if a – b is divisible by 5. Prove that R is an equivalence relation.
  3. Let A = {a, b, c}. Write all possible equivalence relations on A.
  4. A relation R is defined on ℕ by aRb if a divides b. Prove whether R is reflexive, symmetric, or transitive.
Case Study Question

Section D: Case Study Based Question (5 marks)

Case Study:

In a school, the Mathematics teacher is forming groups for a project on “Real-life Applications of Relations.” The students are divided into three groups: Group A, Group B, and Group C. Each group contains 3 students. The teacher defines a relation R such that a pair (x, y) ∈ R if student x and student y are in the same group.

After constructing the relation, she observes that each student is related to themselves, if x is related to y, then y is also related to x, and if x is related to y and y to z, then x is also related to z.

She asks students to observe the properties of the relation R and identify its type and characteristics.

Based on the above case study, answer the following:
  1. The relation R defined above is:
    • (A) Reflexive only
    • (B) Symmetric only
    • (C) Transitive only
    • (D) Equivalence
  2. Number of pairs in R if each group has 3 students:
    • (A) 9
    • (B) 27
    • (C) 3
    • (D) 6
  3. Which of the following is definitely true?
    • (A) (x,x) ∈ R
    • (B) (x,y) ∈ R implies (y,x) ∉ R
    • (C) (x,y) ∈ R implies x = y
    • (D) (x,y) ∈ R implies x > y
  4. The real-life relation defined here models:
    • (A) Division relation
    • (B) Group membership
    • (C) Friendship
    • (D) Ranking
  5. Which property is not violated in R?
    • (A) Symmetry
    • (B) Reflexivity
    • (C) Transitivity
    • (D) None of these

Answer Key

Question Answer
1C
2C
3A
4C
5A
6B
7C
8D
9Yes
10Yes
11{(1,3), (2,3)}, cardinality = 2
12{(1,1), (1,2), (1,3), (2,2), (2,3), (3,3)}
13Yes
14Yes
15Reflexive: Yes, Symmetric: Yes, Transitive: Yes
16Proved as equivalence relation
175 equivalence relations possible
18Reflexive: Yes, Symmetric: No, Transitive: Yes
19D
20B
21A
22B
23D

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