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Chapter 3: Problem 36

Write the negation of each statement. \(\sim p \vee q\)

Short Answer

Expert verified
The negation of the statement \(\sim p \vee q\) is \(p \wedge \sim q\).

Step by step solution

01

Identify the Components of the Statement

Our statement is \(\sim p \vee q\). Here, there are two components: \(\sim p\) and \(q\), separated by an OR operator \(\vee\).
02

Apply De Morgan's Law

To negate a disjunction (OR statement), we must change the OR to AND and negate each component. This gives us \(p \wedge \sim q\).
03

Interpret the Results

Our negated statement is \(p \wedge \sim q\). This represents logical conjunction (AND) of \(p\) and the negation of \(q\).

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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 we are to have peace, we must not encourage the competitive spirit. If we are to make progress, we must encourage the competitive spirit. \(\therefore\) We do not have peace and we do not make progress.

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

Write an original argument in words for the direct reasoning form.

Use Euler diagrams to determine whether each argument is valid or invalid. All actors are artists. Sean Penn is an actor. Therefore, Sean Penn is an artist.

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.
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