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Chapter 3: Problem 68

Determine the truth value for each statement when \(p\) is false, \(q\) is true, and \(r\) is false. \(\sim p \rightarrow q\)

Short Answer

Expert verified
The truth value of the statement \(\sim p \rightarrow q\) when \(p\) is false and \(q\) is true is true.

Step by step solution

01

Evaluate \(\sim p\)

Given that \(p\) is false, \(\sim p\) (not \(p\)), is the opposite of \(p\), so \(\sim p\) is true.
02

Evaluate \( \sim p \rightarrow q \)

Now that you know \(\sim p\) is true, you can evaluate the entire statement \(\sim p \rightarrow q\). The rule for implication is that \(a \rightarrow b\) is false if and only if \(a\) is true and \(b\) is false. In this case, \(a = \sim p\) is true, and \(b = q\) is true. So the statement \( \sim p \rightarrow q \) is true, because it is not the case of \(a\) being true and \(b\) being false.

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Most popular questions from this chapter

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.

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

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

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.

Explain how to use Euler diagrams to determine whether or not an argument is valid.
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