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Chapter 3: Problem 39

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

Short Answer

Expert verified
The truth table for \((r \wedge \sim p) \vee \sim q\) will consist of eight rows due to the three variables considered. The truth values for each of the rows are determined by sequentially applying the logical operators as instructed in the step-by-step process.

Step by step solution

01

Setting up basic truth values

A truth table for three variables (in this case, the variables are p, q, r) should consist of eight rows, implying all possible combinations of truth values for the three variables: true (T) and false (F). So set up a table that has columns for every variable, and write out every combination of T and Fs for those variables: (T,T,T), (T,T,F), (T,F,T)...etc., until all combinations are covered.
02

Compute \(\sim p\) and \(\sim q\)

\(\sim p\) and \(\sim q\) represent the negation of p and q respectively. This can be achieved by flipping the truth values in the p, q column. If a value is T, it becomes F and if it was F, it becomes T.
03

Compute \(r \wedge \sim p\)

The \(r \wedge \sim p\) term represents the conjunction (AND operation) of r and \(\sim p\). In a conjunction, the result is T if and only if both values are T. Otherwise, it's F. So, in this case, when both r and \(\sim p\) are T, the result is T. Else, it should be denoted by F.
04

Compute \((r \wedge \sim p) \vee \sim q\)

The \((r \wedge \sim p) \vee \sim q\) term stands for the disjunction (OR operation) of \(r \wedge \sim p\) and \(\sim q\). In a disjunction, the result is T if at least one of the values is T. Therefore, when either \(r \wedge \sim p\) is T or \(\sim q\) is T, the result is T. If both are F, then it is F.

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Most popular questions from this chapter

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

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 an original argument in words for the contrapositive reasoning form.

Use Euler diagrams to determine whether each argument is valid or invalid. All humans are warm-blooded. No reptiles are human. Therefore, no reptiles are warm-blooded.

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'm at the beach, then I swim in the ocean. If I swim in the ocean, then I feel refreshed. \(\therefore\) If I'm not at the beach, then I don't feel refreshed.
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