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web.groovymark@gmail.com
- November 8, 2024
Question 41
What is the difference between a walk and a trail in graph theory?
a) A walk can repeat vertices but not edges, while a trail cannot repeat edges
b) A walk cannot repeat vertices, while a trail can repeat vertices
c) A walk has no restrictions, while a trail must be closed
d) There is no difference; they are the same
Correct Answer: a) A walk can repeat vertices but not edges, while a trail cannot repeat edges
Explanation: In graph theory, a walk allows for the repetition of vertices and edges, whereas a trail restricts repetition of edges only.
Question 42
What is a proof by contradiction?
a) Proving by directly stating a theorem
b) Assuming the theorem is true and then showing it is false
c) Assuming the theorem is false and showing a contradiction arises
d) Using examples to prove a theorem
Correct Answer: c) Assuming the theorem is false and showing a contradiction arises
Explanation: Proof by contradiction involves starting with the assumption that a statement is false and then demonstrating that this leads to a contradiction, thus proving the original statement must be true.
Question 43
Which of the following is a characteristic of a partial order?
a) It must be symmetric
b) It must be reflexive, transitive, and anti-symmetric
c) It must include every element in the set
d) It must have at least one pair of related elements
Correct Answer: b) It must be reflexive, transitive, and anti-symmetric
Explanation: A partial order requires the relation to be reflexive, transitive, and anti-symmetric, allowing for elements to be compared in a specific manner.
Question 44
What does it mean if a relation R is transitive?
a) If xRy and yRz, then xRz must also hold
b) If xRy, then yRx must hold
c) Every element is related to itself
d) No elements are related to themselves
Correct Answer: a) 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 be related to the third.
Question 45
What is the main use of proofs by cases in mathematics?
a) To simplify complex proofs
b) To prove a predicate is true on its domain by breaking it into classes
c) To show a contradiction
d) To determine the truth of a single statement
Correct Answer: b) To prove a predicate is true on its domain by breaking it into classes
Explanation: Proofs by cases involve dividing a statement into distinct cases or scenarios and proving each one separately, ensuring the overall truth of the statement.
Question 46
What do we call the set of all ordered pairs formed from two sets A and B?
a) Cartesian product
b) Union
c) Intersection
d) Power set
Correct Answer: a) Cartesian product
Explanation: The Cartesian product of two sets A and B is defined as the set of all possible ordered pairs (a,b) where aaa is from A and b is from B.
Question 47
In Boolean algebra, what does the operation + represent?
a) AND
b) OR
c) NOT
d) XOR
Correct Answer: b) OR
Explanation: In Boolean algebra, the + operation signifies the logical OR operation, indicating that at least one of the operands is true.
Question 48
What is the significance of the identity laws in Boolean algebra?
a) They determine equivalence
b) They show that adding or multiplying by the identity does not change the value
c) They prove contradictions
d) They simplify complex equations
Correct Answer: b) They show that adding or multiplying by the identity does not change the value
Explanation: Identity laws state that when you add or multiply a variable by the identity element (0 for addition, 1 for multiplication), the original variable remains unchanged.
Question 49
What is the significance of the double negation law in Boolean algebra?
a) It allows simplification of expressions
b) It states that a double negation results in the original value
c) It defines equivalence
d) It introduces contradiction
Correct Answer: b) It states that a double negation results in the original value
Explanation: The double negation law in Boolean algebra asserts that negating a negation of a variable results in the original variable, thereby simplifying expressions.
Question 50
What does a proof by contraposition typically involve?
a) Proving the original statement directly
b) Proving the contrapositive of the statement
c) Showing a counterexample
d) Breaking the proof into cases
Correct Answer: b) Proving the contrapositive of the statement
Explanation: Proof by contraposition involves demonstrating the truth of a statement by proving that if the conclusion is false, then the premise must also be false.
