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Chapter 3: Problem 35

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

Short Answer

Expert verified
According to the final truth table, the statement \((p \wedge \sim q) \vee(p \wedge q)\) is true when p is true, regardless of the truth value of q, and false when p is false.

Step by step solution

01

Construct the basic table

Let's begin by constructing a table with four permutations of truth values for p and q. It can be represented as follows: \[ \begin{array}{|c|c|c|c|c|} \hline p & q & \sim q & p \wedge \sim q & p \wedge q \ \hline T & T & F & F & T\ T & F & T & T & F\ F & T & F & F & F\ F & F & T & F & F\ \hline \end{array} \]
02

Evaluate \(p \wedge \sim q\) and \(p \wedge q\)

Using the basic table, calculate the values of the intermediate expressions \(p \wedge \sim q\) and \(p \wedge q\). Notice that \(p \wedge \sim q\) is true only when p is true and q is false. On the other hand, \(p \wedge q\) is true only when both p and q are true.
03

Evaluate entire statement

Lastly, compute the overall expression \((p \wedge \sim q) \vee (p \wedge q)\) which is true when either \(p \wedge \sim q\) is true or \(p \wedge q\) is true. Thus, the complete truth table is: \[ \begin{array}{|c|c|c|c|c|c|} \hline p & q & \sim q & p \wedge \sim q & p \wedge q & (p \wedge \sim q) \vee (p \wedge q)\ \hline T & T & F & F & T & T\ T & F & T & T & F & T\ F & T & F & F & F & F\ F & F & T & F & F & F\ \hline \end{array} \]

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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 I tell you I cheated, I'm miserable. If I don't tell you I cheated, I'm miserable. \(\therefore\) I'm miserable.

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 watch Schindler's List and Milk, I am aware of the destructive nature of intolerance. Today I did not watch Schindler's List or I did not watch Milk. \(\therefore\) Today I am not aware of the destructive nature of intolerance.

Exercises 59-60 illustrate arguments that have appeared in cartoons. Each argument is restated below the cartoon. Translate the argument into symbolic form and then determine whether it is valid or invalid. If you do not know how to read, you cannot read War and Peace. If you cannot read War and Peace, then Leo Tolstoy will hate you. Therefore, if you do not know how to read, Leo Tolstoy will hate you.

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 an argument is in the form of the fallacy of the inverse, then it is invalid. This argument is invalid. \(\therefore\) This argument is in the form of the fallacy of the inverse.

Determine whether each argument is valid or invalid. No \(A\) are \(B\), no \(B\) are \(C\), and no \(C\) are \(D\). Thus, no \(A\) are \(D\).
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