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Chapter 3: Problem 73

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

Short Answer

Expert verified
The truth value for the logical statement is true.

Step by step solution

01

Evaluate Inner Statements

Find the values of \(p \rightarrow \sim r\) and \(r \wedge \sim p\). Using the fact that \(p\) is false, \(q\) is true, and \(r\) is false, we find that \(p \rightarrow \sim r\) becomes 'false \rightarrow true', which evaluates to true based on the truth table of '->'. Similarly, \(r \wedge \sim p\) becomes 'false and true', which evaluates to false based on the truth table of 'and'.
02

Evaluate Outer Statement

Substitute the values obtained in step 1 into the outer statement \(\sim[(p \rightarrow \sim r) \leftrightarrow (r \wedge \sim p)]\). This becomes \(\sim[ true \leftrightarrow false]\). The phrase inside the brackets evaluates to false according to the truth table of '\(\leftrightarrow\)'. Hence, the expression becomes \(\sim false\).
03

Final Evaluation

The final step is to compute \(\sim false\). In logic, the 'not' operation negates the truth value. Hence, \(\sim false\) becomes true.

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Most popular questions from this chapter

If you are given an argument in words that contains two premises and a conclusion, describe how to determine if the argument is valid or invalid.

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.) There must be a dam or there is flooding. This year there is flooding. \(\therefore\) This year there is no dam.

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."

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can't use Euler diagrams to determine the validity of an argument if one of the premises is false.

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 I am tired or hungry, I cannot concentrate. I can concentrate. \(\therefore\) I am neither tired nor hungry.
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