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web.groovymark@gmail.com
- November 12, 2024
Question 01
Which of the following describes the principle of inclusion-exclusion?
a) ∣A∪B∣=∣A∣+∣B∣
b) ∣A∩B∣=∣A∣+∣B∣−∣A∪B∣
c) ∣A∪B∣=∣A∣+∣B∣−∣A∩B∣
d) ∣A∪B∣=∣A∣−∣B∣
Correct Answer: c) ∣A∪B∣=∣A∣+∣B∣−∣A∩B∣
Explanation: The principle of inclusion-exclusion provides a formula for calculating the size of the union of two sets, accounting for the overlap (intersection) between them.
Question 02
What does the pigeonhole principle state?
a) If there are more items than containers, at least one container must be empty
b) If there are more items than containers, at least one container must contain more than one item
c) If there are fewer items than containers, all containers will have one item
d) If there are fewer items than containers, at least one container must be empty
Correct Answer: b) If there are more items than containers, at least one container must contain more than one item
Explanation: The pigeonhole principle states that if more items are placed into containers than there are containers, at least one container must contain more than one item.
Question 03
Which of the following describes a permutation of a set?
a) An ordered arrangement of elements
b) An unordered collection of elements
c) A subset of the set
d) A combination of elements
Correct Answer: a) An ordered arrangement of elements
Explanation: A permutation is an ordered arrangement of elements from a set, where the order matters.
Question 04
What is the difference between an r-permutation and an r-combination?
a) A permutation is ordered, but a combination is unordered
b) A permutation is unordered, but a combination is ordered
c) A permutation includes all elements, while a combination excludes some elements
d) A permutation is always larger than a combination
Correct Answer: a) A permutation is ordered, but a combination is unordered
Explanation: In a permutation, the order of elements matters, while in a combination, the order does not matter.
Question 05
Which of the following is an example of the multiplication rule?
a) There are n ways to complete the first task and m ways to complete the second task, so there are nm ways to complete both tasks
b) There are n ways to complete both tasks
c) There are n+m ways to complete both tasks
d) There are n×m2 ways to complete both tasks
Correct Answer: a) There are n ways to complete the first task and mmm ways to complete the second task, so there are nmnmnm ways to complete both tasks
Explanation: The multiplication rule states that if one task can be completed in n ways and another in mmm ways, then both tasks can be completed in nmnmnm ways.
Question 06
What does the generalized pigeonhole principle state?
a) If n objects are placed into k containers, at least one container must contain more than n/k objects
b) If n objects are placed into k containers, all containers will have exactly n/k objects
c) If n objects are placed into k containers, at least one container must contain fewer than n/k objects
d) If n objects are placed into k containers, no container will contain more than n/k objects
Correct Answer: a) If n objects are placed into k containers, at least one container must contain more than n/k objects
Explanation: The generalized pigeonhole principle states that if n objects are placed into k containers, at least one container must contain more than [n/k] objects.
Question 07
Which of the following describes an injective function?
a) A function where every element of the domain maps to a unique element in the codomain
b) A function where every element of the codomain maps to a unique element in the domain
c) A function where every element of the domain maps to the same element in the codomain
d) A function where some elements of the domain map to multiple elements in the codomain
Correct Answer: a) A function where every element of the domain maps to a unique element in the codomain
Explanation: An injective function is one where distinct elements in the domain map to distinct elements in the codomain.
Question 08
Which of the following best describes the set of natural numbers N?
a) The set of all positive and negative integers
b) The set of all integers greater than or equal to zero
c) The set of all rational numbers
d) The set of all real numbers
Correct Answer: b) The set of all integers greater than or equal to zero
Explanation: The set of natural numbers N consists of all non-negative integers, including zero.
Question 09
Which of the following best describes the set of rational numbers Q?
a) The set of numbers that can be expressed as a fraction a/b, where a and b are integers and b≠0
b) The set of all integers
c) The set of all real numbers
d) The set of all complex numbers
Correct Answer: a) The set of numbers that can be expressed as a fraction a/b, where a and b are integers and b≠0
Explanation: The set of rational numbers Q includes all numbers that can be expressed as a ratio of two integers.
Question 10
Which of the following describes the cardinality of a set?
a) The number of elements in the set
b) The sum of the elements in the set
c) The product of the elements in the set
d) The number of distinct subsets of the set
Correct Answer: a) The number of elements in the set
Explanation: The cardinality of a set is the number of elements contained within the set.
Question 11
What is the power set of a set a?
a) The set of all possible subsets of a
b) The set of all elements in a
c) The set of all proper subsets of a
d) The set of all disjoint subsets of a
Correct Answer: a) The set of all possible subsets of a
Explanation: The power set of a set a is the set of all subsets of a, including the empty set and a itself.
Question 12
Which of the following is true for disjoint sets?
a) Their intersection is the empty set
b) Their union is the empty set
c) They contain all the same elements
d) Their difference is the empty set
Correct Answer: a) Their intersection is the empty set
Explanation: Two sets are disjoint if their intersection is the empty set, meaning they have no elements in common.
Question 13
Which of the following describes the complement of a set A in a universal set U?
a) The set of all elements in U that are not in A
b) The set of all elements in A that are not in U
c) The set of all elements in A
d) The set of all elements in U
Correct Answer: a) The set of all elements in U that are not in A
Explanation: The complement of A set a is the set of all elements in the universal set U that are not in A.
Question 14
Which of the following best describes a function?
a) A relation where each element of the domain is paired with exactly one element of the codomain
b) A relation where each element of the codomain is paired with exactly one element of the domain
c) A relation where each element of the domain is paired with multiple elements of the codomain
d) A relation where each element of the codomain is paired with multiple elements of the domain
Correct Answer: a) A relation where each element of the domain is paired with exactly one element of the codomain
Explanation: A function is a relation that assigns exactly one element of the codomain to each element of the domain.
Question 15
Which of the following describes an equivalence relation?
a) A relation that is reflexive, symmetric, and transitive
b) A relation that is antisymmetric and transitive
c) A relation that is reflexive and antisymmetric
d) A relation that is symmetric and antisymmetric
Correct Answer: a) A relation that is reflexive, symmetric, and transitive
Explanation: An equivalence relation is one that satisfies the properties of being reflexive, symmetric, and transitive.
Question 16
Which of the following is an example of a set difference?
a) The set of elements in A that are not in B
b) The set of elements in both A and B
c) The set of elements in A and B
d) The set of all elements in B that are not in A
Correct Answer: a) The set of elements in A that are not in B
Explanation: The set difference A−B is the set of elements that are in A but not in B.
Question 17
Which of the following is true for an injective function?
a) It maps distinct elements of the domain to distinct elements of the codomain
b) It maps distinct elements of the codomain to distinct elements of the domain
c) It maps the same element of the domain to multiple elements of the codomain
d) It maps the same element of the codomain to multiple elements of the domain
Correct Answer: a) It maps distinct elements of the domain to distinct elements of the codomain
Explanation: An injective function ensures that each distinct element in the domain maps to a unique element in the codomain.
Question 18
What does the notation A×B represent in set theory?
a) The Cartesian product of sets A and B
b) The intersection of sets A and B
c) The union of sets A and B
d) The difference between sets A and B
Correct Answer: a) The Cartesian product of sets A and B
Explanation: The Cartesian product A×B represents the set of ordered pairs (a,b) where a∈A and b∈B.
Question 19
Which of the following is a property of a binary relation on a set?
a) Reflexivity
b) Symmetry
c) Transitivity
d) All of the above
Correct Answer: d) All of the above
Explanation: A binary relation on a set can have various properties, including reflexivity, symmetry, and transitivity.
Question 20
Which of the following describes a reflexive relation?
a) Every element in the set is related to itself
b) No elements in the set are related to themselves
c) Only some elements in the set are related to themselves
d) Every element in the set is related to every other element
Correct Answer: a) Every element in the set is related to itself
Explanation: A reflexive relation is one where every element in the set is related to itself.
