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Chapter 3: Problem 58

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

Short Answer

Expert verified
The boolean value for the expression \(\sim(p \vee q) \wedge r\) when 'p' is false, 'q' is true, and 'r' is false is False.

Step by step solution

01

Recognize the Variables and Their Values

Identify the variables and their corresponding values. 'p' is false, 'q' is true, and 'r' is false.
02

Substitute the Boolean Values

Substitute the boolean values of 'p', 'q', and 'r' into the boolean expression. So, the expression becomes: \(\sim(\text{false} \vee \text{true}) \wedge \text{false}\).
03

Apply the OR Operator

The operator inside the parenthesis is OR. When the OR operator is applied to a pair of boolean values, the result is true if at least one of the operands is true. So, false OR true equals to true. The expression, then, simplifies to: \(\sim(\text{true}) \wedge \text{false}\).
04

Apply the NOT Operator

Now apply the NOT operator. The NOT operation negates the boolean value it is applied to. So, NOT true results in false. This simplifies the expression to: \(\text{false} \wedge \text{false}\).
05

Apply the AND Operator

The AND operation gives true if both operands are true, otherwise it gives false. So, False AND False equals to false.

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Most popular questions from this chapter

Use a truth table to determine whether the symbolic form of the argument is valid or invalid. $$ \begin{aligned} &(p \rightarrow q) \wedge(q \rightarrow p) \\ &\frac{p}{\therefore p \vee q} \end{aligned} $$

Write an original argument in words for the direct reasoning form.

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) If it is hot and humid, I complain. It is not hot or it is not humid. \(\therefore\) I am not complaining.

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) If we close the door, then there is less noise. There is less noise. \(\therefore\) We closed the door.

Draw what you believe is a valid conclusion in the form of a disjunction for the following argument. Then verify that the argument is valid for your conclusion. "Inevitably, the use of the placebo involved built-in contradictions. A good patient-doctor relationship is essential to the process, but what happens to that relationship when one of the partners conceals important information from the other? If the doctor tells the truth, he destroys the base on which the placebo rests. If he doesn't tell the truth, he jeopardizes a relationship built on trust."
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