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web.groovymark@gmail.com
- November 8, 2024
Question 01
Which operation is used to combine elements of sets A and B such that every element in A is included along with every element in B?
a) Intersection
b) Symmetric difference
c) Union
d) Difference
Correct Answer: c) Union
Explanation: The union operation combines all elements from both sets A and B, resulting in a set that includes every element from both sets, with no duplicates.
Question 02
What is the proper notation for the power set of a set A?
a) A^C
b) P(A)
c) A*
d) A^P
Correct Answer: b) P(A)
Explanation: The power set of a set A, denoted P(A), is the set of all possible subsets of A, including the empty set and A itself.
Question 03
What does it mean for two sets to be disjoint?
a) They share at least one element
b) They have no elements in common
c) They are equal
d) One is a subset of the other
Correct Answer: b) They have no elements in common
Explanation: Two sets are considered disjoint if their intersection is empty, meaning they do not share any elements.
Question 04
In set theory, what is the complement of set A?
a) All elements in A
b) All elements in the universal set that are not in A
c) Only the empty set
d) Only the elements that are in A but not in B
Correct Answer: b) All elements in the universal set that are not in A
Explanation: The complement of set A includes all elements in the universal set that do not belong to A, effectively representing everything outside of A.
Question 05
What is the notation for the intersection of sets A and B?
a) A + B
b) A ∩ B
c) A * B
d) A – B
Correct Answer: b) A ∩ B
Explanation: The intersection of sets A and B is represented by A∩B and includes only those elements that are present in both A and B.
Question 06
What is a characteristic of a symmetric relation?
a) If xRy, then yRx must also hold
b) Every element relates to itself
c) It has no loops
d) It is both reflexive and transitive
Correct Answer: a) If xRy, then yRx must also hold
Explanation: A relation is symmetric if, whenever one element is related to another, the reverse relationship also holds true.
Question 07
What does it mean for a relation to be anti-reflexive?
a) Every element is related to itself
b) No elements relate to themselves
c) Some elements are unrelated
d) All pairs are related
Correct Answer: b) No elements relate to themselves
Explanation: A relation is anti-reflexive if no element in the set has a self-loop, meaning that for every x, xRx does not hold.
Question 08
Which of the following defines a strict order?
a) Reflexive, transitive, and symmetric
b) Reflexive, transitive, and anti-symmetric
c) Transitive and anti-reflexive
d) Reflexive and symmetric
Correct Answer: c) Transitive and anti-reflexive
Explanation: A strict order is characterized by being transitive (if xRy and yRz, then xRz) and anti-reflexive (no element relates to itself).
Question 09
What does a reflexive relation require?
a) No pairs can relate
b) Every element must relate to at least one other
c) Every element must relate to itself
d) All elements must be different
Correct Answer: c) Every element must relate to itself
Explanation: A relation is reflexive if every element in the set has a self-loop, meaning xRx holds true for all x in the set.
Question 10
How is the number of edges pointing out of a vertex described?
a) Indegree
b) Outdegree
c) Total degree
d) Path length
Correct Answer: b) Outdegree
Explanation: The outdegree of a vertex refers to the number of directed edges that originate from that vertex.
Question 11
What distinguishes a cycle from a circuit in graph theory?
a) A cycle can have repeated edges
b) A circuit can have repeated vertices
c) A cycle cannot repeat vertices except for the starting and ending vertex
d) There is no difference
Correct Answer: c) A cycle cannot repeat vertices except for the starting and ending vertex
Explanation: A cycle is defined as a closed path where no vertices are repeated, except for the starting and ending vertex.
Question 12
Which operation is used to prove a statement by assuming the opposite of the conclusion is true?
a) Direct proof
b) Proof by cases
c) Proof by contradiction
d) Inductive proof
Correct Answer: c) Proof by contradiction
Explanation: Proof by contradiction involves assuming that the conclusion is false and demonstrating that this leads to a logical inconsistency, thus proving the original statement.
Question 13
What type of function is both injective and surjective?
a) Constant function
b) Bijective function
c) Linear function
d) Discrete function
Correct Answer: b) Bijective function
Explanation: A bijective function is defined as a function that is both injective (one-to-one) and surjective (onto), ensuring a perfect pairing between elements of the domain and codomain.
Question 14
What does a proof by exhaustion involve?
a) Assuming a theorem is false
b) Testing each element in the domain
c) Showing that all pairs of elements relate
d) Demonstrating a contradiction
Correct Answer: b) Testing each element in the domain
Explanation: Proof by exhaustion entails checking every individual element within the domain to confirm that the statement holds true for all possible cases.
Question 15
Which of the following describes a path in graph theory?
a) A sequence of edges where no edge occurs more than once
b) A sequence of vertices with repeated edges
c) A sequence of vertices with repeated vertices
d) A sequence of vertices with no restrictions
Correct Answer: a) A sequence of edges where no edge occurs more than once
Explanation: A path is characterized by traversing through vertices without repeating any edges, distinguishing it from other types of walks in graph theory.
Question 16
What does the term “complement” refer to in set theory?
a) The overlap between two sets
b) The total count of elements in a set
c) Elements in the universal set not in a specific set
d) The intersection of two sets
Correct Answer: c) Elements in the universal set not in a specific set
Explanation: The complement of a set includes all elements in the universal set that are not part of that specific set.
Question 17
What is the characteristic of a bijective function?
a) It maps every element of the domain to multiple elements of the codomain
b) It maps distinct elements of the domain to the same element of the codomain
c) It maps every element of the domain to a unique element of the codomain and covers all elements of the codomain
d) It does not have an inverse
Correct Answer: c) It maps every element of the domain to a unique element of the codomain and covers all elements of the codomain
Explanation: A bijective function is both one-to-one and onto, meaning it creates a unique pairing between the domain and the codomain, allowing for an inverse function.
Question 18
What is an equivalence class?
a) The set of all elements related to a single element
b) The intersection of two sets
c) The union of all equivalence relations
d) The power set of a set
Correct Answer: a) The set of all elements related to a single element
Explanation: An equivalence class consists of all elements in the domain that are equivalent to a particular element under a given equivalence relation.
Question 19
What is the primary use of De Morgan’s laws in logic?
a) To simplify expressions
b) To express negations
c) To prove contradictions
d) To create new logical relationships
Correct Answer: a) To simplify expressions
Explanation: De Morgan's laws provide rules for distributing negation across conjunctions and disjunctions, allowing for the simplification of logical expressions.
Question 20
Which of the following operations is used to combine sets such that only unique elements are retained?
a) Intersection
b) Union
c) Symmetric difference
d) Cartesian product
Correct Answer: b) Union
Explanation: The union operation combines two sets and retains only unique elements, effectively merging their contents without duplication.
