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Chapter 3: Problem 3

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

Short Answer

Expert verified
The truth table for the statement \(\sim(q \rightarrow p)\) is:\[\begin{array}{|c|c|c|c|}\hlinep & q & q \rightarrow p & \sim(q \rightarrow p) \\hlineT & T & T & F \T & F & T & F \F & T & F & T \F & F & T & F \\hline\end{array}\]

Step by step solution

01

List all Possible Combinations of Truth Values for the Variables

To start, create a table with columns for \(p\), \(q\), \(q \rightarrow p\) and \(\sim(q \rightarrow p)\). Then, list all possible combinations of truth values for \(p\) and \(q\), which are true and false.
02

Determine the Truth Values for the Implication Statement

Next, determine the truth values of the implication statement \(q \rightarrow p\). Remember, this statement is true in all cases, except when \(q\) is true and \(p\) is false. Then, fill in the column under \(q \rightarrow p\) based on these values.
03

Finish the Truth Table

Finally, determine the truth values for the statement \(\sim(q \rightarrow p)\). The NOT operator flips the truth value of the statement it modifies. This means where the \(q \rightarrow p\) column has a true, the \(\sim(q \rightarrow p)\) column will have a false, and vice versa.

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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 am tired or hungry, I cannot concentrate. I can concentrate. \(\therefore\) I am neither tired nor hungry.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used Euler diagrams to determine that an argument is valid, but when I reverse one of the premises and the conclusion, this new argument is invalid.

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.

Write a valid argument on one of the following questions. If you can, write valid arguments on both sides. a. Should the death penalty be abolished? b. Should Roe v. Wade be overturned? c. Are online classes a good idea? d. Should marijuana be legalized? e. Should grades be abolished? f. Should same-sex marriage be legalized?

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