Chapter Exam

Chapter 1: Relations and Functions — Online MCQ Test

Q: A relation R from set A to set B is defined as:

Answer: A. A subset of A × B — By definition in NCERT, a relation from set A to set B is any subset of the Cartesian product A × B.

Q: If A = {1, 2, 3} and B = {4, 5}, then the number of elements in A × B is:

Answer: B. 6 — The Cartesian product A × B has |A| × |B| = 3 × 2 = 6 elements.

Q: A function f: A → B is defined as a relation where:

Answer: B. Every element of A has exactly one image in B — A function requires that each element in the domain has exactly one corresponding element in the codomain.

Q: For the relation R = {(1, 2), (2, 3), (1, 3)} on A = {1, 2, 3}, which property does R satisfy?

Answer: C. Transitive — R is transitive because (1, 2) ∈ R and (2, 3) ∈ R implies (1, 3) ∈ R. It's not reflexive or symmetric.

Q: If f: A → B and g: B → C are functions, for (g ∘ f)⁻¹ to exist, which condition must hold?

Answer: C. Both f and g must be bijective — For the composition to have an inverse, both functions must be bijective so that (g ∘ f)⁻¹ = f⁻¹ ∘ g⁻¹ exists.

MATHS · CLASS 12 SECOND PUC · Karnataka State Board

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Chapter 1: Relations and Functions — Important Questions & Answers (FAQ)

Frequently asked questions from Karnataka State Board CLASS 12 SECOND PUC MATHS — Chapter 1: Relations and Functions, with answers and explanations. These are sample questions; the exam has its own separate question set.

A relation R from set A to set B is defined as:

A. A subset of A × B ✓
B. A subset of A ∪ B
C. A subset of A ∩ B
D. Equal to A × B

Answer: A. A subset of A × B
By definition in NCERT, a relation from set A to set B is any subset of the Cartesian product A × B.

If A = {1, 2, 3} and B = {4, 5}, then the number of elements in A × B is:

A. 5
B. 6 ✓
C. 8
D. 10

Answer: B. 6
The Cartesian product A × B has |A| × |B| = 3 × 2 = 6 elements.

A function f: A → B is defined as a relation where:

A. Every element of A has at least one image in B
B. Every element of A has exactly one image in B ✓
C. Every element of B has a pre-image in A
D. Some elements of A may not have images in B

Answer: B. Every element of A has exactly one image in B
A function requires that each element in the domain has exactly one corresponding element in the codomain.

For the relation R = {(1, 2), (2, 3), (1, 3)} on A = {1, 2, 3}, which property does R satisfy?

A. Reflexive
B. Symmetric
C. Transitive ✓
D. Equivalence

Answer: C. Transitive
R is transitive because (1, 2) ∈ R and (2, 3) ∈ R implies (1, 3) ∈ R. It's not reflexive or symmetric.

If f: A → B and g: B → C are functions, for (g ∘ f)⁻¹ to exist, which condition must hold?

A. Both f and g must be injective
B. Both f and g must be surjective
C. Both f and g must be bijective ✓
D. f must be injective and g must be surjective

Answer: C. Both f and g must be bijective
For the composition to have an inverse, both functions must be bijective so that (g ∘ f)⁻¹ = f⁻¹ ∘ g⁻¹ exists.