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Chapter 3: Problem 5

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

Short Answer

Expert verified
The truth table for the logical proposition \((p \wedge q) \rightarrow (p \vee q)\) is such that the proposition is true for all possible combinations of truth values for \(p\) and \(q\), i.e., 'True True', 'True False', 'False True', and 'False False'.

Step by step solution

01

Setup - Create Structure for the Truth Table

Begin by drawing a table with columns for \(p\), \(q\), \(p \wedge q\), \(p \vee q\), and \((p \wedge q) \rightarrow (p \vee q)\). List all possible combinations for truth values of \(p\) and \(q\), which are 'True True', 'True False', 'False True' and 'False False'.
02

Evaluate - Determine Values for Intermediary Operators

Evaluate the values for \(p \wedge q\) and \(p \vee q\) for each combination of \(p\) and \(q\). Remember that \(p \wedge q\) is true if and only if both \(p\) and \(q\) are true, and \(p \vee q\) is true if either \(p\) or \(q\) (or both) are true.
03

Evaluate - Determine Values for the Implies Operator

Finally, evaluate the values for \((p \wedge q) \rightarrow (p \vee q)\) for each combination of \(p\) and \(q\). Remember that an implication \((p \rightarrow q)\) is false only when \(p\) is true and \(q\) is false. In all other cases, it is true.

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Most popular questions from this chapter

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

Conservative commentator Rush Limbaugh directed this passage at liberals and the way they think about crime. Of course, liberals will argue that these actions [contemporary youth crime] can be laid at the foot of socioeconomic inequities, or poverty. However, the Great Depression caused a level of poverty unknown to exist in America today, and yet I have been unable to find any accounts of crime waves sweeping our large cities. Let the liberals chew on that. (See, I Told You So, p. 83) Limbaugh's passage can be expressed in the form of an argument: If poverty causes crime, then crime waves would have swept American cities during the Great Depression. Crime waves did not sweep American cities during the Great Depression. \(\therefore\) Poverty does not cause crime. (Liberals are wrong.) Translate this argument into symbolic form and determine whether it is valid or invalid.

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

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 Tim and Janet play, then the team wins. Tim played and the team did not win. \(\therefore\) Janet did not play.

Use Euler diagrams to determine whether each argument is valid or invalid. All dancers are athletes. Savion Glover is an athlete. Therefore, Savion Glover is a dancer.
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