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Chapter 3: Problem 34

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

Short Answer

Expert verified
The truth table for the statement \((p \wedge \sim q) \vee (\sim p \wedge q)\) is as follows: \[ \begin{array}{ccccccc} p & q & \sim p & \sim q & p \wedge \sim q & \sim p \wedge q & (p \wedge \sim q) \vee (\sim p \wedge q) \\ \hline T & T & F & F & F & F & F \\ T & F & F & T & T & F & T \\ F & T & T & F & F & T & T \\ F & F & T & T & F & F & F \end{array} \]

Step by step solution

01

Constructing the Base Table

First, create a base table with columns for \(p\), \(q\), \(\sim p\), \(\sim q\), \(p \wedge \sim q\), \(\sim p \wedge q\), and \((p \wedge \sim q) \vee (\sim p \wedge q)\). Then, enumerate the possible truth values for the basic propositions \(p\) and \(q\). There are \(2^2 = 4\) possible combinations of truth values as there are two variables involved.
02

Filling in Negations

Next, fill in the values for \(\sim p\) and \(\sim q\) using the NOT operation rule. If a proposition is true, its negation is false, and vice versa.
03

Filling in Conjunctions

Now fill in the values for \(p \wedge \sim q\) and \(\sim p \wedge q\) using the AND operation rule. A conjunction is true if and only if both its components are true. All other cases result in a false.
04

Filling in the Final Column

Lastly, complete the final column i.e., \( (p \wedge \sim q) \vee (\sim p \wedge q) \) by applying the OR operation rule. An OR operation is true if at least one of its component propostions is true, which is the case in the constructed truth 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 The Graduate and Midnight Cowboy are shown, then the performance is sold out. Midnight Cowboy was shown and the performance was not sold out. \(\therefore\) The Graduate was not shown.

Use Euler diagrams to determine whether each argument is valid or invalid. All insects have six legs. No spiders have six legs. Therefore, no spiders are insects.

Draw a valid conclusion from the given premises. Then use a truth table to verify your answer. If you only spoke when spoken to and I only spoke when spoken to, then nobody would ever say anything. Some people do say things. Therefore, ...

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.

Write an original argument in words that has a true conclusion, yet is invalid.
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