Question

Let:

P: 3 is an odd number.
Q: 23 is a prime number.
R: 9 is an even number.
State each of the following in words, and determine the truth value of each.
A) P OR R (P V R): True
B) P AND R (P ˄ R): False
C) Q IFF R (Q ⇔ R): True
D) P IMPLIES Q (P ⇒ Q): True
E) R IMPLIES NOT Q (R ⇒ ¬Q): False
F) NOT Q IMPLIES NOT P (¬Q ⇒ ¬P): True

Answer 1

The logical statements involving propositions P, Q, and R were evaluated, yielding various truth values: P OR R is True, P AND R is False, Q IFF R is False, P IMPLIES Q is True, R IMPLIES NOT Q is True, and NOT Q IMPLIES NOT P is False.

Step-by-step explanation:

A student asked about the truth values of several logical statements relating to three propositions: P (3 is an odd number), Q (23 is a prime number), and R (9 is an even number). Let's evaluate each statement:

• P OR R (P ∨ R): The statement '3 is an odd number OR 9 is an even number.' Since 3 is indeed an odd number, the statement is True.
• P AND R (P ∧ R): The statement '3 is an odd number AND 9 is an even number.' Since 9 is not even, the statement is False.
• Q IFF R (Q ⇔ R): The statement '23 is a prime number IF AND ONLY IF 9 is an even number.' Since 23 is a prime but 9 is not even, the bi-conditional statement is False.
• P IMPLIES Q (P ⇒ Q): The statement 'If 3 is an odd number, then 23 is a prime number.' Both propositions are true, making the implication True.
• R IMPLIES NOT Q (R ⇒ ¬Q): The statement 'If 9 is an even number, then 23 is not a prime number.' This is a True implication because when the antecedent is False, the implication is True despite the truth value of the consequent.
• NOT Q IMPLIES NOT P (¬Q ⇒ ¬P): The statement 'If 23 is not a prime number, then 3 is not an odd number.' This implication is False because Q is true, meaning ~Q is false, and thus, anything that implies from a false statement is True.