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web.groovymark@gmail.com
- November 12, 2024
Question 41
Which of the following describes a surjective function?
a) A function where every element in the codomain has at least one preimage
b) A function where every element in the domain has at least one image
c) A function where no two elements in the domain map to the same element
d) A function where no elements in the codomain have preimages
Correct Answer: a) A function where every element in the codomain has at least one preimage
Explanation: A surjective function is one where every element in the codomain is the image of at least one element from the domain.
Question 42
Which of the following is an example of a function that is both injective and surjective?
a) A bijection
b) A permutation
c) A matrix
d) A set
Correct Answer: a) A bijection
Explanation: A bijection is a function that is both injective (one-to-one) and surjective (onto).
Question 43
Which of the following describes a binary relation from set A to set B?
a) A subset of A×B
b) A subset of A×A
c) A subset of B×B
d) A subset of B×A
Correct Answer: a) A subset of A×B
Explanation: A binary relation from set A to set B is a subset of the Cartesian product A×B, where elements from A are related to elements from B.
Question 44
Which of the following is an example of a reflexive relation?
a) (a,a)∈A for all a∈A
b) (a,b)∈A∈A and (b,a)∈A
c) (a,b)∈A, but (b,a)∉A
d) (a,b)∉A for all a,b∈A
Correct Answer: a) (a,a)∈A for all a∈A
Explanation: A reflexive relation requires that every element in the set is related to itself, meaning (a,a)∈A for all a∈A.
Question 45
Which of the following is an example of an antisymmetric relation?
a) If a is related to b and b is related to a, then a=b
b) If a is related to b, then b is also related to a
c) If a is related to b, then b is not related to a
d) If a is related to b, then a is also related to c
Correct Answer: a) If a is related to b and b is related to a, then a=b
Explanation: An antisymmetric relation is one where if a is related to b and b is related to a, then a must equal b.
Question 46
Which of the following is an example of a transitive relation?
a) If a is related to b and b is related to c, then a is related to c
b) If a is related to b, then b is also related to a
c) If a is related to b, then a is equal to b
d) If a is related to b, then a is not related to c
Correct Answer: a) If a is related to b and b is related to c, then a is related to c
Explanation: A transitive relation is one where if one element is related to a second and the second is related to a third, the first is related to the third.
Question 47
Which of the following is an example of an equivalence class?
a) The set of elements that are related to a given element by an equivalence relation
b) The set of elements that are not related to any other elements
c) The set of all elements in a binary relation
d) The set of all elements that are greater than a given element
Correct Answer: a) The set of elements that are related to a given element by an equivalence relation
Explanation: An equivalence class is the set of elements that are related to a given element by an equivalence relation.
Question 48
Which of the following describes a partition of a set?
a) A collection of subsets of the set that are pairwise disjoint and whose union is the original set
b) A collection of subsets of the set that are not disjoint
c) A collection of subsets of the set that contains all possible elements
d) A collection of subsets of the set that contains no elements
Correct Answer: a) A collection of subsets of the set that are pairwise disjoint and whose union is the original set
Explanation: A partition of a set divides the set into non-overlapping subsets whose union equals the original set.
Question 49
Which of the following is an example of a function that is not injective?
a) A function where multiple elements in the domain map to the same element in the codomain
b) A function where each element in the domain maps to a unique element in the codomain
c) A function where every element in the codomain has at least one preimage
d) A function where no elements in the codomain have preimages
Correct Answer: a) A function where multiple elements in the domain map to the same element in the codomain
Explanation: A function is not injective if two or more distinct elements in the domain map to the same element in the codomain.
Question 50
Which of the following is true for the composition of two functions?
a) The composition of two injective functions is injective
b) The composition of two surjective functions is surjective
c) The composition of two bijective functions is bijective
d) All of the above
Correct Answer: d) All of the above
Explanation: The composition of two injective functions is injective, the composition of two surjective functions is surjective, and the composition of two bijective functions is bijective.
