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web.groovymark@gmail.com
- November 9, 2024
Question 01
What is the primary characteristic of a binary relation?
a) It contains only single elements
b) It consists of ordered pairs
c) It is always reflexive
d) It can only relate numbers
Correct Answer: b) It consists of ordered pairs
Explanation: A binary relation is defined as a set that consists solely of ordered pairs, establishing connections between elements of two sets.
Question 02
What does the symbol ¬ represent in logical expressions?
a) AND
b) OR
c) NOT
d) XOR
Correct Answer: c) NOT
Explanation: The symbol ¬ signifies the logical negation operation, indicating the opposite truth value of the statement it is applied to.
Question 03
In graph theory, what does an open walk imply?
a) The first and last vertices are the same
b) The walk can repeat edges
c) The walk cannot repeat any vertices
d) The first and last vertices are different
Correct Answer: d) The first and last vertices are different
Explanation: An open walk is defined by starting and ending at different vertices, allowing for a flexible traversal of edges.
Question 04
What is the notation for the symmetric difference of sets A and B?
a) A ∩ B
b) A ∪ B
c) A Δ B
d) A \ B
Correct Answer: c) A Δ B
Explanation: The symmetric difference of sets A and B, denoted as AΔB, includes elements that are in either A or B but not in both.
Question 05
Which statement is true about the empty set?
a) It contains one element
b) It is a subset of itself
c) It is not a subset of any set
d) It can contain duplicate elements
Correct Answer: b) It is a subset of itself
Explanation: The empty set is considered a subset of itself and every other set, as it contains no elements to violate the subset condition.
Question 06
What does a relation being anti-symmetric imply?
a) Every element is related to itself
b) If xRy and yRx, then x must equal y
c) No elements can relate to themselves
d) All pairs of elements must relate
Correct Answer: b) If xRy and yRx, then x must equal y
Explanation: A relation is anti-symmetric if the presence of both relationships implies that the two elements must be identical.
Question 07
What does the term “reflexive” mean in the context of relations?
a) No elements are related to themselves
b) Every element in the set is related to itself
c) Some elements relate to others
d) It must contain at least one pair of elements
Correct Answer: b) Every element in the set is related to itself
Explanation: A reflexive relation requires that every element in the set has a self-loop, meaning xRx holds for all x.
Question 08
What is the significance of the outdegree of a vertex in a directed graph?
a) It counts incoming edges
b) It counts outgoing edges
c) It counts all edges
d) It indicates isolated vertices
Correct Answer: b) It counts outgoing edges
Explanation: The outdegree of a vertex refers to the number of edges that are directed away from that vertex, indicating its outgoing connections.
Question 09
What does it mean for a function to be bijective?
a) It maps some elements in the domain to multiple elements in the codomain
b) It is not a function
c) It maps every element in the domain to a unique element in the codomain and covers all elements in the codomain
d) It has a range larger than its domain
Correct Answer: c) It maps every element in the domain to a unique element in the codomain and covers all elements in the codomain
Explanation: A bijective function is both injective (one-to-one) and surjective (onto), ensuring that every element is paired uniquely and completely covered.
Question 10
What is a common application of proofs by contradiction?
a) Proving statements are true
b) Finding counterexamples
c) Demonstrating that two functions are equal
d) Establishing that a statement is false by showing an inconsistency
Correct Answer: d) Establishing that a statement is false by showing an inconsistency
Explanation: Proofs by contradiction start by assuming a statement is true and show that this leads to a logical inconsistency, thus proving the statement must be false.
Question 11
What does the notation A⊆B indicate about sets A and B?
a) A is equal to B
b) A is a proper subset of B
c) A is a subset of B
d) B is a subset of A
Correct Answer: c) A is a subset of B
Explanation: The notation A⊆B indicates that every element of set A is also an element of set B, establishing a subset relationship.
Question 12
What is the outcome of A∩A?
a) A
b) B
c) The empty set
d) No outcome
Correct Answer: a) A
Explanation: The intersection of a set A with itself is A, as all elements are common to both sets.
Question 13
What is the relationship between a proper subset and a subset?
a) A proper subset must contain all elements of the set
b) A proper subset must have fewer elements than the set
c) A proper subset is always equal to the set
d) A proper subset can be identical to the set
Correct Answer: b) A proper subset must have fewer elements than the set
Explanation: A proper subset includes at least one element but not all elements of the set, distinguishing it from an improper subset, which can be equal to the original set.
Question 14
What does a proof by cases entail?
a) Proving a single statement
b) Breaking down a statement into distinct scenarios and proving each one
c) Establishing a contradiction
d) Assuming the conclusion is true
Correct Answer: b) Breaking down a statement into distinct scenarios and proving each one
Explanation: Proof by cases involves analyzing different scenarios or conditions that might affect the truth of the statement and proving each one separately.
Question 15
What does the expression A∩∅ equal?
a) A
b) B
c) The empty set
d) The universal set
Correct Answer: c) The empty set
Explanation: The intersection of any set A with the empty set results in the empty set, as there are no elements to share.
Question 16
What is the main property of a function that is not surjective?
a) It maps every element of the domain to a unique element in the codomain
b) At least one element in the codomain does not have a pre-image in the domain
c) It is bijective
d) It maps every element in the codomain to multiple elements in the domain
Correct Answer: b) At least one element in the codomain does not have a pre-image in the domain
Explanation: A function is not surjective if there are elements in the codomain that do not correspond to any elements from the domain, indicating incomplete coverage.
Question 17
What does the term “power set” refer to?
a) The set of all possible elements in a set
b) The set of all subsets of a given set
c) The set of unique elements in a set
d) The set of elements that can be combined
Correct Answer: b) The set of all subsets of a given set
Explanation: The power set is the collection of all possible subsets of a set, including the empty set and the set itself.
Question 18
What does the term “logical equivalence” indicate about two statements?
a) They always yield different truth values
b) They yield the same truth value in all cases
c) They cannot both be true
d) They are independent
Correct Answer: b) They yield the same truth value in all cases
Explanation: Two statements are logically equivalent if they produce identical truth values across all possible interpretations, indicating they convey the same information.
Question 19
What is a characteristic of a tautology in logic?
a) It can be false
b) It is always true
c) It has conditional truth
d) It can only be true in some cases
Correct Answer: b) It is always true
Explanation: A tautology is a logical statement that remains true under all possible truth values for its components.
Question 20
What is the characteristic of a function that is both injective and surjective?
a) It is not a function
b) It is a bijective function
c) It can map one element to multiple elements
d) It has no inverse
Correct Answer: b) It is a bijective function
Explanation: A bijective function ensures a perfect one-to-one correspondence between the elements of the domain and the codomain, making it both injective and surjective.
