Questions: Determine the truth value of the statement (p ∨ ¬q) ∧ r using the following conditions. a) p is true, q is false, and r is true. b) p is false, q is false, and r is false. a) If p is true, q is false, and r is true, what is the truth value of (p ∨ ¬q) ∧ r?

Solution Steps

To determine the truth value of the statement \((p \vee \sim q) \wedge r\) under given conditions, we need to evaluate the logical expression step-by-step for each set of conditions.

1. For condition (a): \(p\) is true, \(q\) is false, and \(r\) is true.

   • Evaluate \(\sim q\) (the negation of \(q\)).
   • Evaluate \(p \vee \sim q\) (the disjunction of \(p\) and \(\sim q\)).
   • Evaluate \((p \vee \sim q) \wedge r\) (the conjunction of the result from the previous step and \(r\)).

2. For condition (b): \(p\) is false, \(q\) is false, and \(r\) is false.

   • Follow the same steps as in condition (a).

For condition (a): \(q\) is false, so \(\sim q\) is true.

For condition (b): \(q\) is false, so \(\sim q\) is true.

For condition (a): \(p\) is true and \(\sim q\) is true, so \(p \vee \sim q\) is true.

For condition (b): \(p\) is false and \(\sim q\) is true, so \(p \vee \sim q\) is true.

For condition (a): \(p \vee \sim q\) is true and \(r\) is true, so \((p \vee \sim q) \wedge r\) is true.

For condition (b): \(p \vee \sim q\) is true but \(r\) is false, so \((p \vee \sim q) \wedge r\) is false.

Final Answer