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Chapter 3: Problem 32

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

Short Answer

Expert verified
Your truth table should have eight rows representing all possible combinations of truth values for \(p\), \(q\), and \(r\), and additional columns for each part of the compound statement. The truth value of the entire statement \([r \wedge (q \vee \sim p)] \leftrightarrow \sim q\) can be determined based on the truth values of its elements using the definitions of the logical operators.

Step by step solution

01

Set up the truth table

Create a table with columns for \(p\), \(q\), \(r\) and each part of the compound statement. Put in all eight combinations of truth values for \(p\), \(q\), and \(r\) (i.e., true (T) or false (F)).
02

Evaluate \(\sim p\)

Evaluate the negation of \(p\) (i.e., \(\sim p\)) for each row of the table by reversing the truth values of \(p\).
03

Evaluate \(q \vee \sim p\)

Evaluate the disjunction of \(q\) and \(\sim p\) (i.e., \(q \vee \sim p\)) for each row of the table. Remember that the disjunction is true if at least one of \(q\) or \(\sim p\) is true, and false otherwise.
04

Evaluate \(r \wedge (q \vee \sim p)\)

Evaluate the conjunction of \(r\) and \(q \vee \sim p\) (i.e., \(r \wedge (q \vee \sim p)\)) for each row of the table. Remember that the conjunction is true if both \(r\) and \(q \vee \sim p\) are true, and false otherwise.
05

Evaluate \(\sim q\)

Evaluate the negation of \(q\) (i.e., \(\sim q\)) for each row of the table by reversing the truth values of \(q\).
06

Evaluate the entire statement

Finally, evaluate the biconditional statement \([r \wedge (q \vee \sim p)] \leftrightarrow \sim q\) for each row of the table. Remember that the biconditional is true if \(r \wedge (q \vee \sim p)\) and \(\sim q\) have the same truth value, and false otherwise.

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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 it is hot and humid, I complain. It is not hot or it is not humid. \(\therefore\) I am not complaining.

Describe what is meant by a valid argument.

Use Euler diagrams to determine whether each argument is valid or invalid. All thefts are immoral acts. Some thefts are justifiable. Therefore, some immoral acts are justifiable.

Use Euler diagrams to determine whether each argument is valid or invalid. All professors are wise people. Some professors are actors. Therefore, some wise people are actors.

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