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Chapter 3: Problem 32

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

Short Answer

Expert verified
The truth table is as follows: (TT):(F,T), (TF):(T,T), (FT):(F,F), (FF):(T,T). Where the first pair in each ordered pair represents \(p\) and \(q\), respectively, and the second part after the colon represents the truth value of \(\sim p\) and \(\sim p \wedge (p \vee \sim q)\), respectively.

Step by step solution

01

List All Possible Combinations of Truth Values for the Variables

There are two variables in the statement: \(p\) and \(q\). As they can only be true or false, there are a total of \(2^2=4\) combinations of values for these variables. The possible combinations are \(TF, TT, FT, FF\), where \(T\) stands for true and \(F\) stands for false.
02

Evaluate \(\sim p\) and \(\sim q\) for Each Combination

\(\sim p\) is the logical negation of \(p\) and \(\sim q\) is the negation of \(q\). This means \(\sim p\) is true whenever \(p\) is false, and false when \(p\) is true. The same applies for \(\sim q\) with respect to \(q\).
03

Evaluate \(p \vee \sim q\) for Each Combination

\(p \vee \sim q\) is the logical 'or' operation. This is true if either \(p\) is true or \(\sim q\) (the negation of \(q\)) is true or both are true. For each combination of truth values for \(p\) and \(q\), if either \(p\) is true or \(q\) is false, then \(p \vee \sim q\) is true.
04

Evaluate \(\sim p \wedge (p \vee \sim q)\) for Each Combination

\(\sim p\wedge(p \vee \sim q)\) is the logical 'and' operation between \(\sim p\) and \(p \vee \sim q\). This is true only if both \(\sim p\) and \(p \vee \sim q\) are true. For each combination of truth values for \(p\) and \(q\), if both \(\sim p\) and \(p \vee \sim q\) are true, then \(\sim p \wedge (p \vee \sim q)\) is true.

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Most popular questions from this chapter

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 .

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can't use Euler diagrams to determine the validity of an argument if one of the premises is false.

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.

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 \(D\).

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.
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