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Chapter 3: Problem 37

Construct a truth table for the given statement. \(p \wedge(\sim q \vee r)\)

Short Answer

Expert verified
The truth table for \(p \wedge(\sim q \vee r)\) will have eight rows due to the three variables present. Each row will correspond to a different configuration of true or false values for \(p\), \(q\), and \(r\). The final output for each row will depend on the logical operations as described in the steps.

Step by step solution

01

Identify Variables

The variables in the given statement are \(p\), \(q\), and \(r\). Create a column for each in the truth table. For each variable, half the entries will be true and the other half will be false. Since we have three variables, our truth table will have \(2^3 = 8\) entries.
02

Compute \(\sim q\)

Calculate the logical 'not' for variable \(q\). If \(q\) is true, then \(\sim q\) is false and vice versa. Add these results into a new column in the truth table.
03

Compute \(\sim q \vee r\)

With the results of \(\sim q\) and \(r\) from the truth table, calculate the logical 'or' between \(\sim q\) and \(r\). If either \(\sim q\) or \(r\) is true, then \(\sim q \vee r\) is true. Add these results into another new column.
04

Compute final expression

Calculate \(p \wedge (\sim q \vee r)\) using the results of column \(p\) and column \(\sim q \vee r\). If both are true, then the result is true. Else, it is false. This operation completes the truth table.

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Most popular questions from this chapter

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 you tell me what I already understand, you do not enlarge my understanding. If you tell me something that I do not understand, then your remarks are unintelligible to me. \(\therefore\) Whatever you tell me does not enlarge my understanding or is unintelligible to me.

Under what circumstances should Euler diagrams rather than truth tables be used to determine whether or not an argument is valid?

Use Euler diagrams to determine whether each argument is valid or invalid. All insects have six legs. No spiders have six legs. Therefore, no spiders are insects.

Use Euler diagrams to determine whether each argument is valid or invalid. All humans are warm-blooded. No reptiles are human. Therefore, no reptiles are warm-blooded.

Use Euler diagrams to determine whether each argument is valid or invalid. All writers appreciate language. All poets are writers. Therefore, all poets appreciate language.
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