Functions and Equivalence Relations - Question Bank

1. Let A be a non-empty set. The relation R = {(a, a) | a ∈ A} on A is:
A) Symmetric and transitive, but not reflexive.
B) Reflexive and symmetric, but not transitive.
C) Reflexive and transitive, but not symmetric.
D) An equivalence relation.
2. If a relation R on set A is reflexive and symmetric, it is called:
A) A partial order
B) A quasi-order
C) A pre-order
D) A total order
3. What is the result of the composition g o f, where f(x) = x² and g(x) = x + 1?
A) (x + 1)²
B) x² + 1
C) x²(x + 1)
D) (x + 1)x²
4. What is the result of the composition f o g, where f(x) = x² and g(x) = x + 1?
A) (x + 1)²
B) x² + 1
C) x²(x + 1)
D) (x + 1)x²
5. Consider the relation 'has the same number of elements as' on the set of all finite sets. This is:
A) A partial order
B) A strict order
C) An equivalence relation
D) A function
6. If f: A -> B and g: B -> A are inverse functions, then f(g(b)) = b for all b in B and g(f(a)) = a for all a in A. This means:
A) f is surjective and g is injective.
B) f is injective and g is surjective.
C) f is bijective and g is bijective.
D) f is reflexive and g is symmetric.
7. The relation R = {(x, y) | x and y are integers and x < y} on the set of integers is:
A) An equivalence relation
B) Reflexive and symmetric
C) Symmetric and transitive
D) Transitive but not reflexive or symmetric
8. The relation R = {(x, y) | x and y are integers and x = y} on the set of integers is:
A) Reflexive only
B) Symmetric only
C) Transitive only
D) An equivalence relation
9. What is the definition of a relation 'a is related to b' (a R b) on a set A?
A) A subset of the Cartesian product A x A.
B) A function from A to A.
C) A subset of A.
D) A single element from A.
10. If f: A -> B is a function and g: B -> C is a function, and if g is not surjective, can g o f be surjective?
A) Yes, always.
B) No, never.
C) Yes, if f is surjective.
D) Yes, if f is injective.
11. If f: A -> B is a function and g: B -> C is a function, and if f is not injective, can g o f be injective?
A) Yes, always.
B) No, never.
C) Yes, if g is injective.
D) Yes, if g is surjective.
12. Consider the function f(x) = x³ - x. Is this function injective?
A) Yes, it is always injective.
B) No, for example, f(1) = 0 and f(-1) = 0.
C) No, because f(0) = 0.
D) Yes, because its range is all real numbers.
13. Consider the function f(x) = |x|. Is this function injective?
A) Yes, because different inputs always give different outputs.
B) No, because f(2) = 2 and f(-2) = 2.
C) Yes, because it is always non-negative.
D) No, because its range is all real numbers.
14. What is the range of the function f(x) = x²?
A) All real numbers.
B) All positive real numbers.
C) All non-negative real numbers.
D) All integers.
15. What is the domain of the function f(x) = sqrt(x)?
A) All real numbers.
B) All positive real numbers.
C) All non-negative real numbers.
D) All integers.
16. For the relation R in the previous question, what are the equivalence classes?
A) {[1, 2], [3, 4]}
B) {[1], [2], [3], [4]}
C) {[1, 3], [2, 4]}
D) {[1, 2, 3, 4]}
17. Let A = {1, 2, 3, 4} and R = {(1,1), (2,2), (3,3), (4,4), (1,2), (2,1), (3,4), (4,3)}. Is R an equivalence relation?
A) No, it's not reflexive.
B) No, it's not symmetric.
C) Yes, it is an equivalence relation.
D) No, it's not transitive.
18. If R is an equivalence relation on A, and a R b, then the equivalence class of a, [a], and the equivalence class of b, [b], satisfy:
A) [a] is a proper subset of [b]
B) [a] is equal to [b]
C) [a] and [b] are disjoint
D) [a] contains [b]
19. Consider the relation '≤' on the set of real numbers. This is an example of:
A) An equivalence relation
B) A partial order relation
C) A strict order relation
D) A relation that is not reflexive
20. A relation that is reflexive, antisymmetric, and transitive is called:
A) An equivalence relation
B) A partial order relation
C) A total order relation
D) A function
21. A relation R on a set A is antisymmetric if:
A) For all a, b in A, if (a, b) is in R and (b, a) is in R, then a = b.
B) For all a in A, (a, a) is in R.
C) For all a, b in A, if (a, b) is in R, then (b, a) is in R.
D) For all a, b in A, if (a, b) is in R, then a != b.
22. What does it mean for a function to be strictly increasing?
A) If x₁ < x₂, then f(x₁) ≤ f(x₂).
B) If x₁ < x₂, then f(x₁) > f(x₂).
C) If x₁ < x₂, then f(x₁) < f(x₂).
D) If x₁ < x₂, then f(x₁) ≥ f(x₂).
23. Which of the following is a property of functions but not necessarily of relations?
A) Reflexivity
B) Symmetry
C) Uniqueness of output for each input
D) Transitivity
24. If f(x) = 2x + 1, what is the inverse function f⁻¹(y)?
A) (y - 1) / 2
B) (y + 1) / 2
C) 2y - 1
D) 1 / (2x + 1)
25. In the context of the relation 'a R b if a - b is even' on integers, what is the equivalence class of the integer 3?
A) The set of all odd integers.
B) The set of all even integers.
C) The set {3}
D) The set of all integers.
26. Consider the set of integers Z and the relation R defined as a R b if a - b is an even number. Is R an equivalence relation?
A) No, it's not reflexive.
B) No, it's not symmetric.
C) No, it's not transitive.
D) Yes, it is an equivalence relation.
27. If f: A -> B is a function, and g: B -> A is its inverse (g = f⁻¹), which property must hold?
A) f o g = id_B and g o f = id_A
B) f o g = id_A and g o f = id_B
C) f o g = id_A only
D) g o f = id_B only
28. What is the identity function, id_A, on a set A?
A) A function that maps every element of A to a single fixed element.
B) A function defined as id_A(x) = x for all x in A.
C) A function that maps every element of A to the empty set.
D) A function that maps every element of A to itself, but only if A has only one element.
29. For a function f: A -> B, when does the inverse relation f⁻¹ exist as a function from B to A?
A) When f is surjective.
B) When f is injective.
C) When f is bijective.
D) When f is reflexive.
30. If f is a bijective function from set A to set B, what can be said about the inverse function f⁻¹?
A) It does not exist.
B) It is a function from B to A.
C) It is a relation but not necessarily a function.
D) It maps every element of B to multiple elements of A.
31. Let A = {1, 2, 3} and R = {(1,1), (2,2), (3,3), (1,2), (2,1)}. Is R an equivalence relation on A?
A) No, it's not reflexive.
B) No, it's not symmetric.
C) No, it's not transitive.
D) Yes, it is an equivalence relation.
32. The set of all equivalence classes of a set A under an equivalence relation R is called:
A) The Cartesian product of A
B) The power set of A
C) The partition of A
D) The relation R itself
33. What is a key property of equivalence classes formed by an equivalence relation?
A) They are always disjoint or identical.
B) They always overlap partially.
C) They are never empty.
D) They always contain exactly one element.
34. If R is an equivalence relation on a set A, the set of all elements related to an element 'a' in A is called:
A) The domain of R
B) The codomain of R
C) The equivalence class of 'a'
D) The range of R
35. Consider the relation 'is congruent to' on the set of all triangles. Is this relation an equivalence relation?
A) Yes, it is reflexive, symmetric, and transitive.
B) No, because it is not reflexive.
C) No, because it is not symmetric.
D) No, because it is not transitive.
36. Consider the relation 'is parallel to' on the set of all lines in a plane. Is this relation an equivalence relation?
A) No, because it is not reflexive.
B) No, because it is not symmetric.
C) Yes, it is reflexive, symmetric, and transitive.
D) No, because it is not transitive.
37. Which property is NOT required for a relation to be an equivalence relation?
A) Reflexivity
B) Symmetry
C) Transitivity
D) Antisymmetry
38. A relation R on a set A is transitive if:
A) For all a in A, (a, a) is in R.
B) For all a, b in A, if (a, b) is in R, then (b, a) is in R.
C) For all a, b, c in A, if (a, b) is in R and (b, c) is in R, then (a, c) is in R.
D) For all a, b in A, if (a, b) is in R, then a != b.
39. A relation R on a set A is symmetric if:
A) For all a in A, (a, a) is in R.
B) For all a, b in A, if (a, b) is in R, then (b, a) is in R.
C) For all a, b, c in A, if (a, b) is in R and (b, c) is in R, then (a, c) is in R.
D) For all a, b in A, (a, b) is in R implies a = b.
40. A relation R on a set A is reflexive if:
A) For all a in A, (a, b) is in R whenever (b, a) is in R.
B) For all a in A, (a, a) is in R.
C) For all a, b in A, if (a, b) is in R and (b, c) is in R, then (a, c) is in R.
D) For all a, b in A, if (a, b) is in R and a is not equal to b, then (b, a) is not in R.
41. What is an equivalence relation?
A) A relation that is reflexive, symmetric, and transitive.
B) A relation that is reflexive, antisymmetric, and transitive.
C) A relation that is symmetric, antisymmetric, and transitive.
D) A relation that is reflexive, symmetric, and intransitive.
42. If f: A -> B and g: B -> C are functions, what is the composition of f and g, denoted by g o f?
A) A function from A to C defined by (g o f)(x) = g(f(x)).
B) A function from B to A defined by (g o f)(x) = f(g(x)).
C) A function from A to C defined by (g o f)(x) = f(g(x)).
D) A function from C to A defined by (g o f)(x) = g(f(x)).
43. A function that is both injective and surjective is called:
A) A constant function
B) A bijective function
C) An identity function
D) A trivial function
44. A function f: A -> B is called injective (or one-to-one) if:
A) Different elements in A map to the same element in B.
B) Different elements in A map to different elements in B.
C) Every element in B is mapped to by at least one element in A.
D) The range of f is equal to the codomain B.
45. A function f: A -> B is called surjective (or onto) if:
A) Every element in A maps to a unique element in B.
B) Every element in B is mapped to by at least one element in A.
C) Every element in A maps to exactly one element in B.
D) The domain A is equal to the codomain B.
46. The range of a function is defined as:
A) The set of all possible input values.
B) The set of all possible output values.
C) The set of all values in the codomain that are actually mapped to.
D) The set of values for which the function is positive.
47. What is the codomain of a function?
A) The set of all actual output values produced by the function.
B) The set of all possible input values.
C) The set of all values that the function's output can potentially take.
D) The set of values where the function is continuous.
48. In the context of functions, what does the domain refer to?
A) The set of all possible output values.
B) The set of all possible input values.
C) The set of all ordered pairs in the relation.
D) The set of values for which the function is undefined.
49. What is the fundamental definition of a function in mathematics?
A) A relation where each input maps to exactly one output.
B) A relation where each output maps to exactly one input.
C) A set of ordered pairs with no restrictions.
D) A rule that assigns multiple outputs to a single input.
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