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web.groovymark@gmail.com
- November 9, 2024
Question 21
What does the expression ¬(p∧q)\neg( p \land q)¬(p∧q) simplify to?
a) p∨q
b) ¬p∧¬q
c) ¬p∨¬q
d) p∨¬q
Correct Answer: c) ¬p∨¬q
Explanation: According to De Morgan's laws, the negation of a conjunction simplifies to the disjunction of the negations.
Question 22
What does the expression A×B denote?
a) The intersection of A and B
b) The union of A and B
c) The Cartesian product of sets A and B
d) The power set of A
Correct Answer: c) The Cartesian product of sets A and B
Explanation: The notation A×B represents the Cartesian product, which consists of all possible ordered pairs (a,b) where a∈A and b∈B.
Question 23
What does the notation P(A) represent in set theory?
a) The intersection of A
b) The union of A
c) The power set of A
d) The complement of A
Correct Answer: c) The power set of A
Explanation: The notation P(A) denotes the power set of A, which includes all possible subsets of the set A.
Question 24
What is the significance of the identity law in Boolean algebra?
a) It states that a variable does not change when combined with its identity element
b) It shows that all variables are equal
c) It provides the basis for negation
d) It defines disjunction
Correct Answer: a) It states that a variable does not change when combined with its identity element
Explanation: The identity law indicates that adding zero to a variable or multiplying it by one yields the original variable, maintaining its value.
Question 25
Which of the following describes a proper subset?
a) It contains all elements of the other set
b) It contains at least one but not all elements of the other set
c) It is equal to the other set
d) It is disjoint from the other set
Correct Answer: b) It contains at least one but not all elements of the other set
Explanation: A proper subset includes some elements of the original set but not all, distinguishing it from an improper subset, which can be equal to the original set.
Question 26
What does the operation A∪B yield?
a) Elements that are in A and B
b) Elements that are in A or B or both
c) Elements that are in neither A nor B
d) The power set of A and B
Correct Answer: b) Elements that are in A or B or both
Explanation: The union operation A∪B combines all elements from both sets, including duplicates only once.
Question 27
What defines a cycle in a directed graph?
a) A closed path with no repeated edges
b) A path that does not repeat vertices
c) A path that starts and ends at the same vertex without repeating any edges
d) A walk that can repeat any edge
Correct Answer: c) A path that starts and ends at the same vertex without repeating any edges
Explanation: A cycle in a directed graph is a closed path where the first and last vertices are the same and no edges are traversed more than once.
Question 28
What does the term “logic gate” refer to in digital circuits?
a) A physical barrier in circuits
b) A device that performs a logical operation on one or more inputs
c) A storage component
d) A type of resistor
Correct Answer: b) A device that performs a logical operation on one or more inputs
Explanation: Logic gates are the fundamental building blocks of digital circuits that carry out basic logical functions such as AND, OR, and NOT.
Question 29
What is the result of A∩AC?
a) A
b) The empty set
c) The universal set
d) B
Correct Answer: b) The empty set
Explanation: The intersection of a set A with its complement AC yields the empty set, as no elements can be both in A and not in A simultaneously.
Question 30
Which of the following describes a counterexample?
a) An example that supports a claim
b) An example that disproves a general statement
c) An irrelevant example
d) An instance of a theorem
Correct Answer: b) An example that disproves a general statement
Explanation: A counterexample serves to demonstrate that a general statement or claim is false by providing a specific case where the statement does not hold.
Question 31
What is the meaning of the term “bounded variable”?
a) A variable that is unrestricted
b) A variable that is quantified
c) A variable with a fixed value
d) A variable that can take on any value
Correct Answer: b) A variable that is quantified
Explanation: A bounded variable is one that is subject to a quantifier, thus limiting its range to a specific set of values.
Question 32
What does the operation A∪(B∩C) simplify to?
a) A
b) A ∩ B
c) A ∪ B ∪ C
d) A ∪ B
Correct Answer: d) A ∪ B
Explanation: The operation simplifies to the union of A with the elements that are both in B and C, effectively including all elements of A and those common to B and C.
Question 33
In terms of functions, what does “surjective” mean?
a) Every element in the domain maps to a unique element in the codomain
b) There are elements in the codomain that do not have pre-images in the domain
c) Every element in the codomain has at least one corresponding element in the domain
d) It maps multiple elements to a single output
Correct Answer: c) Every element in the codomain has at least one corresponding element in the domain
Explanation: A surjective function ensures that all elements of the codomain are represented by at least one element from the domain, guaranteeing complete coverage.
Question 34
What does a proper subset A⊊B imply?
a) A is equal to B
b) A contains all elements of B
c) A contains some elements of B but not all
d) A and B have no elements in common
Correct Answer: c) A contains some elements of B but not all
Explanation: The notation A⊊B signifies that A is a proper subset of B, meaning it includes some but not all elements of B.
Question 35
What does the expression p→q signify?
a) If p is true, then q is false
b) If p is true, then q is also true
c) p and q are equivalent
d) p is false regardless of q
Correct Answer: b) If p is true, then q is also true
Explanation: The expression p→q denotes a conditional statement indicating that the truth of q depends on the truth of p.
Question 36
What does the term “antidomain” refer to in a function?
a) The range of the function
b) The set of all possible outputs
c) The original set of inputs
d) The set of elements that do not map to any outputs
Correct Answer: d) The set of elements that do not map to any outputs
Explanation: The antidomain represents values in the codomain that are not associated with any elements from the domain, indicating outputs that lack pre-images.
Question 37
What is the significance of the double complement law in Boolean algebra?
a) It states that p∨¬p is a tautology
b) It indicates that a double negation yields the original value
c) It defines contradictions
d) It shows the equivalence of AND and OR
Correct Answer: b) It indicates that a double negation yields the original value
Explanation: The double complement law asserts that negating a negation of a variable returns the original variable, simplifying expressions in Boolean logic.
Question 38
Which property is essential for a strict total order?
a) It is reflexive
b) It is symmetric
c) It is transitive and anti-reflexive
d) It can have equal elements
Correct Answer: c) It is transitive and anti-reflexive
Explanation: A strict total order is characterized by being transitive (if xRy and yRz, then xRz) and anti-reflexive (no element relates to itself).
Question 39
What is the role of De Morgan’s laws in simplifying logical expressions?
a) To create new logical expressions
b) To transform disjunctions and conjunctions with negation
c) To prove contradictions
d) To establish equivalences
Correct Answer: b) To transform disjunctions and conjunctions with negation
Explanation: De Morgan's laws provide rules for distributing negation across conjunctions and disjunctions, aiding in the simplification of logical expressions.
Question 40
What does the expression ¬(q∧r) simplify to using De Morgan’s laws?
a) ¬q∧¬r
b) ¬q∨¬r
c) q∨r
d) ¬r∧¬q
Correct Answer: b) ¬q∨¬r
Explanation: According to De Morgan's laws, the negation of a conjunction is equivalent to the disjunction of the negations, hence ¬(q∧r)=¬q∨¬r.
