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Chapter 3: Problem 55

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

Short Answer

Expert verified
The truth value of the statement \(\sim p \vee(q \wedge \sim r)\) under the given truth values for \(p\), \(q\), and \(r\) is True.

Step by step solution

01

Substitute the Truth Values

First, substitute the known truth values for each of the variables in the statement. The statement \(\sim p \vee(q \wedge \sim r)\) becomes \(\sim \text{False} \vee(\text{True} \wedge \sim \text{False})\).
02

Apply the Not Operator

Next, apply the 'not' operator (\(\sim\)) to the variables it precedes. This changes the truth value of these variables to the opposite of what it is. The statement becomes \(\text{True} \vee(\text{True} \wedge \text{True})\).
03

Apply the And Operator

Now, apply the 'and' operator (\(\wedge\)) which results in True if both of the operands are True. In this case, the statement becomes \(\text{True} \vee \text{True}\).
04

Apply the Or Operator

Lastly, apply the 'or' operator (\(\vee\)), which results in True if at least one of the operands is True. The final truth value of the statement is \(\text{True}\).

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