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Chapter 3: Problem 65

Construct a truth table for each statement. \([(p \wedge \sim r) \vee(q \wedge \sim r)] \wedge \sim(\sim p \vee r)\)

Short Answer

Expert verified
The truth table includes eight scenarios depending on the truth values of p, q and r. In each scenario, the logic of \([(p \wedge \sim r) \vee (q \wedge \sim r)] \wedge \sim(\sim p \vee r)\) is evaluated, resulting in either True or False depending upon the scenario. The critical part is to follow the order of operations (i.e., conjunctions, disjunction, negations) and construct the table step by step.

Step by step solution

01

Set up the table

Start by setting up a truth table with columns for p, q, r and each of the intermediary terms in the compound statement. The table should have 8 rows, representing all possible combinations of True (T) and False (F) for p, q, and r.
02

Fill in truth values for basic propositions

Next, fill in the True and False values for p, q, and r. Half of the p column should be True, and half False. Do the same for q and r, but divide the column into quarters (TTFF for True, FFFF for False respectively).
03

Calculate the values of negated terms

Calculate the truth values for the negated terms, \(\sim r\) and \(\sim p\). The truth value of \(\sim r\) and \(\sim p\) will be the opposite of the values of r and p respectively. Fill in these values in the respective columns.
04

Calculate the values for the conjunctions \(p \wedge \sim r\) and \(q \wedge \sim r\)

Fill in the truth values for the conjunctions \(p \wedge \sim r\) and \(q \wedge \sim r\) by finding where both p and \(\sim r\), and q and \(\sim r\) are True respectively.
05

Calculate the value for the disjunction \((p \wedge \sim r) \vee (q \wedge \sim r)\)

Now find the disjunction \((p \wedge \sim r) \vee (q \wedge \sim r)\) and fill that column in by finding where either \(p \wedge \sim r\) or \(q \wedge \sim r\) is True.
06

Calculate the value for \(\sim p \vee r\)

Fill in the truth values for \(\sim p \vee r\) by finding where either \(\sim p\) or r is True.
07

Calculate the values for \(\sim(\sim p \vee r)\)

Calculate the truth values of \(\sim(\sim p \vee r)\). The values will be the opposite of those in \(\sim p \vee r\). Fill these values in the appropriate column.
08

Calculate the final value of the compound statement

Finally, find the truth values for the entire compound statement \([(p \wedge \sim r) \vee (q \wedge \sim r)] \wedge \sim(\sim p \vee r)\) by finding where both \((p \wedge \sim r) \vee (q \wedge \sim r)\) and \(\sim(\sim p \vee r)\) are True. This will be the final column of the 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 he was disloyal, his dismissal was justified. If he was loyal, his dismissial was justified. \(\therefore\) His dismissal was justified.

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 some journalists learn about the invasion, the newspapers will print the news. If the newspapers print the news, the invasion will not be a secret. The invasion was a secret. \(\therefore\) No journalists learned about the invasion.

Use a truth table to determine whether the symbolic form of the argument is valid or invalid. $$ \begin{aligned} &p \rightarrow q \\ &\underline{q \rightarrow r} \\ &\therefore \sim p \rightarrow \sim r \end{aligned} $$

Translate the argument below into symbolic form. Then use a truth table to determine if the argument is valid or invalid. It's wrong to smoke in public if secondary cigarette smoke is a health threat. If secondary cigarette smoke were not a health threat, the American Lung Association would not say that it is. The American Lung Association says that secondary cigarette smoke is a health threat. Therefore, it's wrong to smoke in public.

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