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web.groovymark@gmail.com
- November 9, 2024
Question 21
What defines an injective function?
a) It maps at least two different elements to the same element
b) It maps every element of the domain to a unique element of the codomain
c) It has a range that is smaller than its domain
d) It can have elements that map to multiple outputs
Correct Answer: b) It maps every element of the domain to a unique element of the codomain
Explanation: An injective function, or one-to-one function, ensures that no two distinct elements in the domain map to the same element in the codomain.
Question 22
What is the characteristic of a transitive relation?
a) If xRy holds, then yRx must also hold
b) If xRy and yRz, then xRz must also hold
c) Every element relates to itself
d) No elements are related
Correct Answer: b) If xRy and yRz, then xRz must also hold
Explanation: A relation is transitive if whenever one element is related to a second element and that second element is related to a third, the first element must also relate to the third.
Question 23
What does the term “functional completeness” refer to in Boolean operations?
a) The ability to express any Boolean function using a specific set of operations
b) The condition where all outputs are the same
c) The restriction of operations to a minimum
d) The use of only one type of gate in a circuit
Correct Answer: a) The ability to express any Boolean function using a specific set of operations
Explanation: Functional completeness implies that a particular set of Boolean operations can be used to represent any Boolean expression, allowing for full expressiveness in logic.
Question 24
Which of the following operations is represented by the notation x with bar on top in Boolean algebra?
a) AND
b) OR
c) NOT
d) XOR
Correct Answer: c) NOT
Explanation: The notation xˉ signifies the logical NOT operation, indicating the negation of the variable x.
Question 25
What is the result of a double negation in Boolean algebra?
a) Always true
b) Always false
c) The original value
d) A contradiction
Correct Answer: c) The original value
Explanation: The double negation law states that negating a negation returns the original value, thus simplifying expressions involving double negatives.
Question 26
What describes the process of rewriting a Boolean expression into a more efficient form using only addition and complement operations?
a) Simplification
b) Direct proof
c) Conversion to CNF
d) Indirect proof
Correct Answer: c) Conversion to CNF
Explanation: To convert a Boolean expression into conjunctive normal form (CNF), the expression must be expressed in terms of addition and complements.
Question 27
Which operation is equivalent to “NOT AND” in Boolean logic?
a) AND
b) OR
c) NAND
d) NOR
Correct Answer: c) NAND
Explanation: The NAND operation is defined as the negation of the AND operation, indicating that it produces a true output unless both inputs are true.
Question 28
What is the fundamental property of a Boolean circuit?
a) It can only produce one output
b) It consists of gates that perform logical operations
c) It requires multiple outputs
d) It cannot have inputs
Correct Answer: b) It consists of gates that perform logical operations
Explanation: A Boolean circuit is comprised of interconnected gates that execute logical operations on input values to produce a single output.
Question 29
What is the significance of the universal quantifier in logic?
a) It denotes specific cases
b) It applies to all elements in a domain
c) It specifies a single element
d) It indicates a contradiction
Correct Answer: b) It applies to all elements in a domain
Explanation: The universal quantifier is used to express that a statement is true for every element in the specified domain, denoting a general rule.
Question 30
What is the existential quantifier used for?
a) To show a universal truth
b) To indicate that at least one element satisfies a condition
c) To negate a statement
d) To define an equivalence class
Correct Answer: b) To indicate that at least one element satisfies a condition
Explanation: The existential quantifier expresses that there exists at least one element in the domain for which a given condition holds true.
Question 31
What is the outcome of applying De Morgan’s laws to the expression ¬(p∧q)?
a) ¬p∧¬q
b) ¬p∨¬q
c) p∨q
d) p∧q
Correct Answer: b) ¬p∨¬q
Explanation: De Morgan's laws state that the negation of a conjunction is equivalent to the disjunction of the negations, thus ¬(p∧q)=¬p∨¬q.
Question 32
Which characteristic defines a function as surjective?
a) Every element of the codomain has a unique pre-image
b) Every element of the domain maps to at least one element in the codomain
c) Every element of the codomain has at least one element in the domain mapping to it
d) Every element of the domain has a single unique output
Correct Answer: c) Every element of the codomain has at least one element in the domain mapping to it
Explanation: A function is surjective (onto) if every element in the codomain is the image of at least one element from the domain, ensuring complete coverage of the codomain.
Question 33
What does it mean for a set to be a proper subset of another set?
a) It contains all elements of the other set
b) It contains some but not all elements of the other set
c) It has the same elements as the other set
d) It is empty
Correct Answer: b) It contains some but not all elements of the other set
Explanation: A proper subset includes at least one element of the larger set but does not contain every element, distinguishing it from an improper subset, which can be equal to the larger set.
Question 34
Which of the following operations results in a set containing elements that are in either A or B, but not both?
a) Union
b) Intersection
c) Symmetric difference
d) Cartesian product
Correct Answer: c) Symmetric difference
Explanation: The symmetric difference includes elements that are unique to each set A and B, effectively excluding elements that are present in both sets.
Question 35
What is the result of the operation ¬(p∨q) according to De Morgan’s law?
a) p∧q
b) ¬p∧¬q
c) ¬p∨¬q
d) p∨q
Correct Answer: b) ¬p∧¬q
Explanation: De Morgan's law states that the negation of a disjunction is equivalent to the conjunction of the negations, hence ¬(p∨q)=¬p∧¬q.
Question 36
What does the notation f:X→Y represent?
a) A function from set Y to set X
b) A function from set X to set Y
c) A relation between two sets
d) An equivalence class
Correct Answer: b) A function from set X to set Y
Explanation: The notation f:X→Y indicates that fff is a function that maps elements from set X to elements in set Y.
Question 37
What does the term “bounded variable” refer to in logic?
a) A variable that can take on any value
b) A variable that is quantified
c) A variable with a fixed value
d) A variable with no restrictions
Correct Answer: b) A variable that is quantified
Explanation: A bounded variable is one that is subject to a quantifier, meaning it is restricted to a specific range of values in its expression.
Question 38
What defines a contradiction in logical terms?
a) A statement that is always true
b) A statement that can be both true and false
c) A statement that is always false
d) A statement that depends on the context
Correct Answer: c) A statement that is always false
Explanation: A contradiction is a logical statement that cannot be true under any circumstances, consistently yielding a false outcome.
Question 39
Which of the following best describes a Boolean function?
a) It can only output one value
b) It maps Boolean inputs to a Boolean output
c) It does not require inputs
d) It can only perform addition
Correct Answer: b) It maps Boolean inputs to a Boolean output
Explanation: A Boolean function processes one or more Boolean inputs (true or false) to produce a Boolean output, making it a fundamental element of logic.
Question 40
What does it mean for a statement to be a tautology?
a) It is always false
b) It is sometimes true
c) It is always true
d) It is conditional
Correct Answer: c) It is always true
Explanation: A tautology is a logical statement that holds true in all possible scenarios, regardless of the truth values of its components.
