Expressing a Relation as a Subset of Cartesian Product

1

easyIf A = {1, 2} and B = {3, 4}, how many elements are in A × B?

A. 2 B. 4 C. 6 D. 8

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✔ Correct: B

n(A) = 2, n(B) = 2. n(A × B) = 2 × 2 = 4.

2

easyIf A = {a, b}, how many subsets does A × A have?

A. 4 B. 8 C. 16 D. 32

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✔ Correct: C

n(A × A) = 2 × 2 = 4. Number of subsets = 2⁴ = 16.

3

mediumLet A = {1, 2, 3} and R = {(1, 2), (2, 3)}. What is the domain of R?

A. {1, 2, 3} B. {1, 2} C. {2, 3} D. {1, 3}

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✔ Correct: B

Domain is the set of first components: {1, 2}.

4

mediumIf R = {(x, y) : x + y = 5, x, y ∈ {1, 2, 3, 4}}, find R.

A. {(1, 4), (2, 3), (3, 2), (4, 1)} B. {(1, 4), (2, 3)} C. {(2, 3), (3, 2)} D. {(1, 4), (4, 1)}

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✔ Correct: A

Check all pairs in {1, 2, 3, 4} × {1, 2, 3, 4} that sum to 5.

5

hardIf A = {1, 2} and B = {3, 4}, which of the following is NOT a relation from A to B?

A. {(1, 3)} B. {(1, 3), (2, 4)} C. {(1, 4), (2, 3), (3, 1)} D. {(2, 4)}

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✔ Correct: C

(3, 1) is not in A × B because 3 ∉ A and 1 ∉ B.

6

hardLet A = {1, 2, 3} and B = {4, 5}. How many relations can be defined from A to B?

A. 6 B. 32 C. 64 D. 128

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✔ Correct: C

n(A × B) = 3 × 2 = 6. Number of relations = 2⁶ = 64.

7

very hardIf R = {(x, x²) : x is a prime number less than 10}, find the range of R.

A. {4, 9, 25, 49} B. {2, 3, 5, 7} C. {1, 4, 9, 16} D. {4, 9, 25, 49, 121}

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✔ Correct: A

Primes < 10 are {2, 3, 5, 7}. Squares are {4, 9, 25, 49}.

1

Assertion (A): The relation R = {(1, 2), (2, 3)} is a subset of {1, 2, 3} × {2, 3, 4}.
Reason (R): A relation R from A to B is defined as a subset of A × B.

Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

A. Option A B. Option B C. Option C D. Option D

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✔ Correct: A

Both the assertion and reason are true, and the reason correctly explains the definition.

2

Assertion (A): If A = {1, 2}, the number of relations on A is 4.
Reason (R): The number of relations on a set A is 2 to the power of n(A).

Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

A. Option A B. Option B C. Option C D. Option D

Show Answer

✔ Correct: D

Assertion is false (it is 2⁴ = 16). Reason is also false (it is 2 to the power of n(A)²).

1

mediumGiven A = {1, 2, 3} and B = {4, 5}, find: (i) A × B (ii) B × A (iii) Is A × B = B × A?

Show Answer & Solution

(i) A × B = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)}
(ii) B × A = {(4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)}
(iii) No, because ordered pairs are different.

2

mediumLet R = {(x, y) : y = x + 2, x, y ∈ {1, 2, 3, 4, 5}}. Find: (i) R (ii) Domain (iii) Range

Show Answer & Solution

(i) R = {(1, 3), (2, 4), (3, 5)}
(ii) Domain = {1, 2, 3}
(iii) Range = {3, 4, 5}

3

hardIf A = {1, 2, 3, 4}, define R = {(x, y) : y = 2x}. Find: (i) R (ii) Domain (iii) Range

Show Answer & Solution

(i) R = {(1, 2), (2, 4)}
(ii) Domain = {1, 2}
(iii) Range = {2, 4}

1

hardLet A = {1, 2, 3, 4, 5, 6}. Define a relation R from A to A by R = {(x, y) : y = x + 1}. (i) Write R in roster form. (ii) Find domain. (iii) Find range. (iv) Find codomain. (v) Is (6, 7) ∈ R?

Show Answer & Solution

Step 1: Identify possible x values from A. x can be 1, 2, 3, 4, 5.
Step 2: Calculate y = x + 1 for each x. y = 2, 3, 4, 5, 6.
Step 3: R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}.
Step 4: Domain = {1, 2, 3, 4, 5}, Range = {2, 3, 4, 5, 6}, Codomain = {1, 2, 3, 4, 5, 6}.
Step 5: (6, 7) ∉ R because 7 ∉ A.

2

very hardLet A = {1, 2, 3} and B = {1, 2, 3, 4}. Define R = {(x, y) : x < y}. (i) List all elements of R. (ii) Find n(R). (iii) Find n(A × B). (iv) Verify R ⊂ A × B. (v) How many total relations exist from A to B?

Show Answer & Solution

Step 1: List pairs where x < y: (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4).
Step 2: n(R) = 6.
Step 3: n(A × B) = 3 × 4 = 12.
Step 4: Since all 6 elements of R are in A × B, R ⊂ A × B is verified.
Step 5: Total relations = 2¹² = 4096.

★

Case Study: A group of 5 students A = {Amit, Binu, Chitra, Dev, Esha} and their test scores B = {60, 70, 80, 90}. A relation R is defined such that (student, score) ∈ R if the student scored at least 70. R = {(Amit, 80), (Binu, 70), (Chitra, 90), (Dev, 60), (Esha, 70)}.

Sub-question 1: (i) Write the subset R of A × B. (ii) Find the domain of R.

Show Solution

(i) R = {(Amit, 80), (Binu, 70), (Chitra, 90), (Dev, 60), (Esha, 70)}. (ii) Domain = {Amit, Binu, Chitra, Dev, Esha}.

Sub-question 2: (i) If we define a new relation R' where score > 75, find R'. (ii) Find the range of R'.

Show Solution

(i) R' = {(Amit, 80), (Chitra, 90)}. (ii) Range = {80, 90}.

Cartesian Product of Sets

Definition of a Relation

Domain and Range of a Relation

Section A: Multiple Choice Questions (MCQs)

Section B: Assertion–Reason Questions

Section C: Short Answer Questions

Section D: Long Answer Questions

Section E: Case-Based Question

CBSE Board & Sample Paper Past Year Questions

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