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Chapter 3: Problem 8

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

Short Answer

Expert verified
Here is a simple four-row truth table that represents the given statement: \n \\begin{tabular}{ | c | c | c | c | c | } \n \\hline \n \\(p\) & \(q\) & \(p \rightarrow q\) & \(\sim p\) & \((p \rightarrow q) \wedge \sim p\) \ [0.5ex] \n \\hline \n \T & T & T & F & F \ \hline \n \T & F & F & F & F \ \hline \n \F & T & T & T & T \ \hline \n \F & F & T & T & T \ \hline \n \\end{tabular}

Step by step solution

01

Create Initial Truth Table

Firstly, construct a truth table with columns for \(p\), \(q\), \(p \rightarrow q\) and \(\sim p\). Populate the columns for \(p\) and \(q\) with all possible combinations of truth values i.e., both true, both false and one true while the other false.
02

Calculate Value for \(p \rightarrow q\)

Next, determine the truth values for the column \(p \rightarrow q\). Recall that a conditional statement is only false when the antecedent (\(p\)) is true and the consequent (\(q\)) is false. In all other cases, it is true.
03

Calculate Value for \(\sim p\)

Fill in the column for \(\sim p\) by noting that the negation operation simply flips the truth values of \(p\) - if \(p\) is true, \(\sim p\) is false and vice versa.
04

Calculate Value for the Overall Statement

Finally, determine the truth values for the overall statement \((p \rightarrow q) \wedge \sim p\). This involves the conjunction operation, which is true only when both constituent parts are true.

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Most popular questions from this chapter

In Exercises 15-42, 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 cold, my motorcycle will not start. My motorcycle started. \(\therefore\) It is not cold.

Use Euler diagrams to determine whether each argument is valid or invalid. All multiples of 6 are multiples of 3 . Eight is not a multiple of 3 . Therefore, 8 is not a multiple of 6 .

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.

Use Euler diagrams to determine whether each argument is valid or invalid. All dancers are athletes. Savion Glover is a dancer. Therefore, Savion Glover is an athlete.

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 cannot concentrate. \(\therefore\) I am tired or hungry.
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