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web.groovymark@gmail.com
- November 9, 2024
Question 21
What is the logical expression p∧(q∨r)p \land (q \lor r)p∧(q∨r) equivalent to?
a) (p∧q)∨(p∧r)
b) p∨(q∧r)
c) (p∨q)∧(p∨r)
d) p∧q∧r
Correct Answer: a) (p∧q)∨(p∧r)
Explanation: This expression demonstrates the distributive law in Boolean algebra, indicating that the conjunction distributes over the disjunction.
Question 22
What is the primary purpose of the simplification rule in logic?
a) To prove the truth of a statement
b) To reduce complex statements into simpler forms
c) To demonstrate contradictions
d) To establish equivalence
Correct Answer: b) To reduce complex statements into simpler forms
Explanation: The simplification rule allows for breaking down logical expressions into simpler components, facilitating easier analysis and understanding.
Question 23
What does the notation ¬(p→q) simplify to?
a) p∧¬q
b) ¬p∨q
c) p∨q
d) ¬p∧q
Correct Answer: a) p∧¬q
Explanation: The negation of the implication p→q simplifies to the conjunction of p and the negation of q, indicating that p is true while q is false.
Question 24
What is a primary characteristic of a binary operation?
a) It operates on two elements from the same set
b) It can operate on any number of elements
c) It produces only integer outputs
d) It operates on elements from different sets
Correct Answer: a) It operates on two elements from the same set
Explanation: A binary operation involves combining two elements from the same set to produce another element in that set, adhering to closure.
Question 25
What is the result of A∩U, where U is the universal set?
a) A
b) The empty set
c) U
d) B
Correct Answer: a) A
Explanation: The intersection of any set A with the universal set U will always return A, as all elements of A are included in U.
Question 26
What does the notation A∪BC represent?
a) The elements in A that are not in B
b) The union of A with the complement of B
c) The intersection of A and B
d) The difference between A and B
Correct Answer: b) The union of A with the complement of B
Explanation: The notation A∪BC combines all elements in A with those elements that are not in B, effectively including elements outside B.
Question 27
Which property is essential for a function to be injective?
a) At least one element in the codomain is not covered
b) Each element in the domain maps to a unique element in the codomain
c) It maps multiple elements to the same output
d) It is surjective
Correct Answer: b) Each element in the domain maps to a unique element in the codomain
Explanation: An injective function ensures that no two distinct elements in the domain are mapped to the same element in the codomain, preserving uniqueness.
Question 28
What does the operation A∪∅ yield?
a) A
b) B
c) The empty set
d) The universal set
Correct Answer: a) A
Explanation: The union of any set A with the empty set results in A itself, as the empty set contributes no elements to the union.
Question 29
In logic, what does p∧q signify?
a) Either p or q must be true
b) Both p and q must be true
c) Only p must be true
d) p must be false
Correct Answer: b) Both p and q must be true
Explanation: The expression p∧q indicates a conjunction where both statements must hold true for the entire expression to be true.
Question 30
What does it mean for a statement to be a tautology?
a) It can be false under certain conditions
b) It is always true
c) It is true in some cases
d) It has no truth value
Correct Answer: b) It is always true
Explanation: A tautology is a statement that holds true in all possible scenarios, regardless of the truth values of its components.
Question 31
What does the expression A×BA \times BA×B produce?
a) The union of sets A and B
b) The intersection of sets 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 expression A×B refers to the Cartesian product, which consists of all ordered pairs formed by combining elements of sets A and B.
Question 32
What does the expression AC mean in set theory?
a) The elements of A
b) The union of A and the universal set
c) The complement of A
d) The intersection of A and the empty set
Correct Answer: c) The complement of A
Explanation: The notation AC denotes the complement of set A, including all elements in the universal set that are not in A.
Question 33
What does the symbol ∧represent in logic?
a) OR
b) AND
c) NOT
d) XOR
Correct Answer: b) AND
Explanation: The symbol ∧denotes the logical AND operation, indicating that both operands must be true for the overall expression to be true.
Question 34
What is the logical operation represented by p∨q?
a) Both p and q must be true
b) At least one of p or q must be true
c) p must be true, but q can be false
d) q must be true, but p can be false
Correct Answer: b) At least one of p or q must be true
Explanation: The expression p∨q represents a disjunction, which holds true if at least one of the statements is true.
Question 35
What is the outcome of the operation ¬(p∧q) according to De Morgan’s laws?
a) ¬p∧¬q
b) ¬p∨¬q
c) p∨q
d) ¬p∧q
Correct Answer: b) ¬p∨¬q
Explanation: According to De Morgan's laws, the negation of a conjunction simplifies to the disjunction of the negations, hence ¬(p∧q) = ¬p∨¬q.
Question 36
What does it mean for a function to be non-injective?
a) It maps every element of the domain to a unique element in the codomain
b) It has no outputs
c) At least two different elements in the domain map to the same element in the codomain
d) It can be bijective
Correct Answer: c) At least two different elements in the domain map to the same element in the codomain
Explanation: A function is non-injective if it loses uniqueness in mapping, where different inputs result in the same output.
Question 37
What does a proof by contradiction aim to achieve?
a) To prove the truth of a statement
b) To show the falseness of a statement by demonstrating a contradiction
c) To confirm all hypotheses
d) To illustrate an example
Correct Answer: b) To show the falseness of a statement by demonstrating a contradiction
Explanation: Proof by contradiction starts by assuming a statement is true and shows that this leads to a logical inconsistency, thus proving the statement must be false.
Question 38
What does the expression p∧¬p illustrate?
a) A tautology
b) A contradiction
c) A valid argument
d) An equivalence
Correct Answer: b) A contradiction
Explanation: The expression p∧¬p is inherently contradictory because it states that ppp is both true and false simultaneously, which is impossible.
Question 39
Which property of a function describes it being “onto”?
a) It is injective
b) It covers all elements in the codomain
c) It has no outputs
d) It maps distinct elements to the same element
Correct Answer: b) It covers all elements in the codomain
Explanation: An "onto" function, or surjective function, ensures that every element in the codomain has at least one pre-image from the domain, covering all possible outputs.
Question 40
What is the result of A∪B∪C if A, B, and C are disjoint sets?
a) The empty set
b) A union of B and C
c) A set containing all unique elements from A, B, and C
d) A set containing only elements from A
Correct Answer: c) A set containing all unique elements from A, B, and C
Explanation: The union of disjoint sets A, B, and C includes all unique elements from each set without duplicates.
