Relations and Functions Important Questions.

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1. Q1Easy

   Which of the following relations on the set of integers is **reflexive**?

   • A.R = {(a, b) | a = b + 1}
   • B.R = {(a, b) | a ≤ b}
   • C.R = {(a, b) | a < b}
   • D.R = {(a, b) | a > b}
   Answer: R = {(a, b) | a ≤ b}

   Explanation: A relation R on a set A is reflexive if (a, a) ∈ R for every a ∈ A. For example, the relation R = {(a, b) | a ≤ b} is reflexive because every integer a satisfies a ≤ a.

2. Q2Easy

   If f: ℝ → ℝ is defined by f(x) = x³, is f **injective**?

   • A.No
   • B.Yes
   • C.Cannot be determined
   • D.Depends on the domain and codomain
   Answer: Yes

   Explanation: A function f is injective if f(a) = f(b) implies a = b. For f(x) = x³, if f(a) = f(b), then a³ = b³ ⇒ a = b.

3. Q3Easy

   Which of the following functions from ℝ to ℝ is surjective?

   • A.f(x) = x²
   • B.f(x) = x³
   • C.f(x) = eˣ
   • D.f(x) = |x|
   Answer: f(x) = x³

   Explanation: f(x) = x³ is surjective on ℝ because for any y ∈ ℝ, there exists x = ∛y such that f(x) = y. x² and |x| map to [0,∞) only. eˣ maps to (0,∞) only.

4. Q4Easy

   Which of the following relations is **not symmetric**?

   • A.R = {(a, b) | a = b}
   • B.R = {(a, b) | a ≤ b}
   • C.R = {(a, b) | a < b}
   • D.R = {(a, b) | a + b = 10}
   Answer: R = {(a, b) | a < b}

   Explanation: The relation R = {(a, b) | a < b} is not symmetric because if a < b, it does not imply b < a.

5. Q5Medium

   Which of the following relations on the set of integers is **reflexive**?

   • A.(a) R = {(x, y) | x + y = 5}
   • B.(b) R = {(x, y) | x < y}
   • C.(c) R = {(x, x) | x ∈ ℤ}
   • D.(d) R = {(x, y) | x > y}
   Answer: (c) R = {(x, x) | x ∈ ℤ}

   Explanation: A relation is reflexive if (a, a) ∈ R for every a in the set. This ensures every element is related to itself.

6. Q6Medium

   Consider the function f: ℝ → ℝ defined by f(x) = x³. Which of the following statements is **true**?

   • A.(a) The function is injective but not surjective.
   • B.(b) The function is surjective but not injective.
   • C.(c) The function is both injective and surjective.
   • D.(d) The function is neither injective nor surjective.
   Answer: (c) The function is both injective and surjective.

   Explanation: A function is bijective if it is both injective (one-to-one) and surjective (onto). f(x) = x³ is strictly increasing, ensuring distinct inputs produce distinct outputs (injective) and every real number is mapped (surjective).

7. Q7Medium

   Which of the following relations on the set A = {1, 2, 3} is **symmetric**?

   • A.(a) R = {(1, 2), (2, 3)}
   • B.(b) R = {(1, 1), (2, 2), (3, 3)}
   • C.(c) R = {(1, 2), (2, 1), (2, 3), (3, 2)}
   • D.(d) R = {(1, 2), (2, 3), (3, 1)}
   Answer: (c) R = {(1, 2), (2, 1), (2, 3), (3, 2)}

   Explanation: A relation is symmetric if (a, b) ∈ R implies (b, a) ∈ R. Here, every pair (a, b) has its reverse (b, a) included.

8. Q8Medium

   If f: X → Y is a function and Y is finite, which of the following is **not necessarily true**?

   • A.(a) f is surjective if |X| ≥ |Y|.
   • B.(b) f is injective if |X| ≤ |Y|.
   • C.(c) f is injective.
   • D.(d) f is surjective if |X| = |Y|.
   Answer: (c) f is injective.

   Explanation: For finite sets, a function being injective does not guarantee surjectivity unless the cardinalities of X and Y are equal. However, injectivity alone is not guaranteed if the domain is larger than the codomain.

9. Q9Hard

   Which of the following relations is an **equivalence relation**?

   • A.R = {(a, b) | a < b}
   • B.R = {(a, b) | a = b + 1}
   • C.R = {(a, b) | a ≡ b mod 3}
   • D.R = {(a, b) | a divides b}
   Answer: R = {(a, b) | a ≡ b mod 3}

   Explanation: An equivalence relation is reflexive, symmetric, and transitive. For R = {(a, b) | a ≡ b mod 3}, all three properties hold: (a, a) ∈ R (reflexive), (a, b) ∈ R ⇒ (b, a) ∈ R (symmetric), and (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R (transitive).

10. Q10Hard

    Which of the following relations on the set of integers is not transitive?

    • A.R = {(a, b) | a ≤ b}
    • B.R = {(a, b) | a divides b}
    • C.R = {(a, b) | a is the father of b}
    • D.R = {(a, b) | a = b}
    Answer: R = {(a, b) | a is the father of b}

    Explanation: The relation 'is the father of' is not transitive because if a is the father of b and b is the father of c, then a is the grandfather of c, not the father. All other options (≤, divides, =) are transitive.

11. Q11Hard

    Consider the relation R on the set of integers defined by R = {(a, b) | a + b is even}. Is R an **equivalence relation**?

    • A.No
    • B.Yes
    • C.Cannot be determined
    • D.Depends on the set
    Answer: Yes

    Explanation: R is reflexive because a + a = 2a is always even. R is symmetric because if a + b is even, then b + a is also even. R is transitive because if a + b and b + c are even, then a + c is even.

12. Q12Hard

    Which of the following functions is **neither injective nor surjective** on ℝ?

    • A.f(x) = x³
    • B.f(x) = e^x
    • C.f(x) = x³ - 3x
    • D.f(x) = sin(x)
    Answer: f(x) = sin(x)

    Explanation: The function f(x) = sin(x) is not injective because sin(0) = sin(π) = 0 with 0 ≠ π. It is not surjective on ℝ because its range is [-1, 1], not all real numbers.

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