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web.groovymark@gmail.com
- November 10, 2024
Question 41
Which of the following best describes the principle of strong mathematical induction?
a) The statement must be proven true for all values of n at once
b) The statement must be true for the base case and hold for all k before proving it for k+1
c) The statement must be true for k+1 in order to prove it for all values of k
d) The statement must be true for even numbers to be true for odd numbers
Correct Answer: b) The statement must be true for the base case and hold for all k before proving it for k+1
Explanation: Strong mathematical induction requires proving the statement for the base case and showing that if it holds for all values up to k, it also holds for k+1.
Question 42
Which of the following is true for an r-permutation of a set?
a) It is an unordered selection of r elements from the set
b) It is an ordered arrangement of r elements from the set
c) It is an unordered arrangement of all elements from the set
d) It is an ordered arrangement of all elements from the set
Correct Answer: b) It is an ordered arrangement of r elements from the set
Explanation: An r-permutation is an ordered arrangement of r elements chosen from a set.
Question 43
What is the difference between a permutation and a combination?
a) A permutation involves order, while a combination does not
b) A combination involves order, while a permutation does not
c) Both permutations and combinations involve order
d) Neither permutations nor combinations involve order
Correct Answer: a) A permutation involves order, while a combination does not
Explanation: A permutation considers the order of elements, while a combination does not.
Question 44
What does the inclusion-exclusion principle calculate?
a) The number of elements in the union of two or more sets
b) The number of elements in the intersection of two or more sets
c) The number of elements in a Cartesian product
d) The number of elements in the complement of a set
Correct Answer: a) The number of elements in the union of two or more sets
Explanation: The inclusion-exclusion principle is used to calculate the size of the union of two or more sets, accounting for any overlap.
Question 45
Which of the following statements is true for a binary relation on set A?
a) It relates elements of set A to elements of set B
b) It relates elements of set A to themselves
c) It relates every element of set A to every other element of set A
d) It relates elements of set A to elements of set A
Correct Answer: d) It relates elements of set A to elements of set A
Explanation: A binary relation on set A is a subset of A×A, meaning it relates elements of set A to other elements in the same set.
Question 46
Which of the following describes a reflexive relation?
a) If a is related to b, then b is related to a
b) Every element in the set is related to itself
c) No element is related to any other element
d) Every element is related to every other element
Correct Answer: b) Every element in the set is related to itself
Explanation: A reflexive relation means that each element in the set is related to itself.
Question 47
Which of the following describes a symmetric relation?
a) If a is related to b, then b must also be related to a
b) If a is related to b, then a must be greater than b
c) If a is related to b, then a must be equal to b
d) If a is related to b, then a must not be related to b
Correct Answer: a) If a is related to b, then b must also be related to a
Explanation: A symmetric relation ensures that if an element is related to another, the second element must also be related to the first.
Question 48
Which of the following is true for a transitive relation?
a) If a is related to b and b is related to c, then a must be related to c
b) If a is related to b, then b must also be related to a
c) If a is related to b, then a must not be related to b
d) If a is related to b, then a must be equal to b
Correct Answer: a) If a is related to b and b is related to c, then a must be related to c
Explanation: A transitive relation means that if one element is related to a second element, and the second element is related to a third, then the first element is also related to the third.
Question 49
What is a strict order?
a) A relation that is transitive and anti-reflexive
b) A relation that is reflexive and symmetric
c) A relation that is transitive and symmetric
d) A relation that is reflexive and transitive
Correct Answer: a) A relation that is transitive and anti-reflexive
Explanation: A strict order is a relation that is both transitive and anti-reflexive, meaning no element is related to itself.
Question 50
What is a directed acyclic graph (DAG)?
a) A directed graph with no positive length cycles
b) A graph where every vertex is connected to every other vertex
c) A graph where all edges are undirected
d) A graph with exactly one cycle
Correct Answer: a) A directed graph with no positive length cycles
Explanation: A directed acyclic graph (DAG) is a directed graph that contains no cycles of positive length.
