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Ch-1: Relations and Functions
12 Questions
Before You Practice Try each question on your own first. Use the Show Answer button only after you have selected your option, then read the explanation and revise the related concept from your chapter notes.
1. A relation R on a set A is called reflexive if:
A. (a, a) ∈ R for every a ∈ A
B. (a, b) ∈ R implies (b, a) ∈ R
C. (a, b), (b, c) ∈ R implies (a, c) ∈ R
D. R is an empty set
Show Answer
Answer: A. (a, a) ∈ R for every a ∈ A
Explanation: A reflexive relation requires every element of A to be related to itself.
2. A relation R is called symmetric if:
A. (a, b) ∈ R implies (b, a) ∈ R
B. (a, a) ∈ R for all a
C. (a, b), (b, c) ∈ R implies (a, c) ∈ R
D. None of these
Show Answer
Answer: A. (a, b) ∈ R implies (b, a) ∈ R
Explanation: Symmetry requires that whenever a is related to b, b must also be related to a.
3. An equivalence relation must be:
A. Reflexive, symmetric and transitive
B. Only reflexive
C. Only symmetric
D. Only transitive
Show Answer
Answer: A. Reflexive, symmetric and transitive
Explanation: An equivalence relation satisfies all three properties simultaneously: reflexivity, symmetry, and transitivity.
4. A function f: A → B is one-one (injective) if:
A. Distinct elements of A always have distinct images in B
B. Every element of B has a preimage
C. f(a) = f(b) never holds even for a = b
D. A and B have the same cardinality
Show Answer
Answer: A. Distinct elements of A always have distinct images in B
Explanation: Injectivity means no two different elements of the domain map to the same element of the codomain.
5. A function f: A → B is onto (surjective) if:
A. Every element of B has at least one preimage in A
B. f is one-one
C. A and B are equal sets
D. f(a) = a for all a
Show Answer
Answer: A. Every element of B has at least one preimage in A
Explanation: Surjectivity requires the range of f to be the entire codomain B.
6. The number of equivalence relations possible on a finite set is directly related to the:
A. Number of partitions of the set (Bell numbers)
B. Factorial of the number of elements
C. Square of the number of elements
D. Power set size only
Show Answer
Answer: A. Number of partitions of the set (Bell numbers)
Explanation: Each equivalence relation on a set corresponds exactly to a partition of that set into disjoint subsets.
7. A bijective function is one which is:
A. Both one-one and onto
B. Only onto
C. Only one-one
D. Neither one-one nor onto
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Answer: A. Both one-one and onto
Explanation: A bijection is a function that is simultaneously injective and surjective.
8. If f: R → R is defined by f(x) = x², then f is:
A. Neither one-one nor onto
B. One-one only
C. Onto only
D. Bijective
Show Answer
Answer: A. Neither one-one nor onto
Explanation: f(x) = x² is not one-one since f(2) = f(-2), and it is not onto since negative reals have no preimage.
9. The relation R = {(a, b) : a ≤ b} defined on the set of real numbers is:
A. Reflexive and transitive but not symmetric
B. Symmetric only
C. An equivalence relation
D. Neither reflexive nor transitive
Show Answer
Answer: A. Reflexive and transitive but not symmetric
Explanation: a ≤ a holds (reflexive) and a ≤ b, b ≤ c implies a ≤ c (transitive), but a ≤ b does not imply b ≤ a, so it is not symmetric.
10. The composition of two functions (f∘g)(x) means:
A. f(g(x))
B. g(f(x))
C. f(x) · g(x)
D. f(x) + g(x)
Show Answer
Answer: A. f(g(x))
Explanation: In composition, g is applied first and its output is then used as the input to f.
11. If a function f: A → B is invertible, then f must necessarily be:
A. Bijective
B. Only injective
C. Only surjective
D. Any arbitrary function
Show Answer
Answer: A. Bijective
Explanation: A function has an inverse if and only if it is both one-one and onto.
12. The identity function f(x) = x defined on a set A is:
A. Bijective
B. Only injective
C. Only surjective
D. Neither injective nor surjective
Show Answer
Answer: A. Bijective
Explanation: The identity function maps every element to itself, so it is trivially both one-one and onto.
