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Chapter 3: Problem 64

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

Short Answer

Expert verified
The truth table for the statement \(\sim[\sim(p \vee \sim q) \wedge \sim(\sim p \wedge q)]\) is four rows by eight columns, reflecting all combinations of truth values for \(p\) and \(q\) and the resulting truth values of the various subexpressions and the full expression.

Step by step solution

01

Set up the basic truth table

Start the truth table with two variables \(p\) and \(q\), and list all possible combinations of truth values for \(p\) and \(q\). There are \(2^2 = 4\) combinations since we have two variables.
02

Evaluate the small expressions

Evaluate \(\sim q\), \((p \vee \sim q)\), \(\sim p\), and \((\sim p \wedge q)\), and add these columns to the truth table. The value for \(\sim q\) is the negation of the value of \(q\). Similarly, \(\sim p\) is the negation of \(p\). \((p \vee \sim q)\) is true if at least one of \(p\) and \(\sim q\) is true. Similarly, \((\sim p \wedge q)\) is true only if both \(\sim p\) and \(q\) are true.
03

Evaluate the larger expressions

Now evaluate \(\sim (p \vee \sim q)\) and \(\sim (\sim p \wedge q)\) and add these columns to the truth table. These expressions take the negation of the respective expressions in step 2.
04

Evaluate the final expression

Evaluate the final expression \(\sim[\sim(p \vee \sim q) \wedge \sim(\sim p \wedge q)]\) and add this column to the truth table. This expression is the negation of the AND of the expressions in step 3.

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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. $$ \begin{aligned} &p \leftrightarrow q \\ &\underline{q \rightarrow r} \\ &\therefore \sim r \rightarrow \sim p \end{aligned} $$

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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.

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Use the standard forms of valid arguments to draw a valid conclusion from the given premises. If all houses meet the hurricane code, then none of them are destroyed by a category 4 hurricane. Some houses were destroyed by Andrew, a category 4 hurricane. Therefore, ...
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