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Chapter 3: Problem 60

Determine the truth value for each statement when \(p\) is false, \(q\) is true, and \(r\) is false. \(\sim(p \wedge q) \vee \sim(p \vee r)\)

Short Answer

Expert verified
The truth value for the statement \(\sim(p \wedge q) \vee \sim(p \vee r)\) when \(p\) is false, \(q\) is true, and \(r\) is false is True.

Step by step solution

01

Evaluate Logical Operations within Parentheses

According to the problem, \(p\) is false, \(q\) is true, and \(r\) is false. Start by evaluating the formulas inside the brackets. \(\sim(p \wedge q)\) becomes \(\sim(F \wedge T)\), which performs a logical AND on False and True. The result of this will be False. \(\sim(p \vee r)\) becomes \(\sim(F \vee F)\) which performs a logical OR on False and False. The result of this will be False.
02

Apply Logical NOT Operator

Now, apply the logical NOT (\(\sim\)) operator. This operator will flip the truth value of the given value. So \(\sim(F \wedge T)\) becomes \(\sim F\) or True, and \(\sim(F \vee F)\) becomes \(\sim F\) or True.
03

Evaluate Logical OR Operator

Finally, evaluate the logical OR (\(\vee\)) operator on the results from step 2. We are now left with \(T \vee T\), which performs a logical OR on True and True. The result of this operation will be True.

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