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Chapter 3: Problem 42

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

Short Answer

Expert verified
A truth table for the logical statement \(\sim(p \wedge q) \vee \sim r\) will have eight rows, representing all possible truth values for \(p\), \(q\), and \(r\). The final column for the entire expression will be computed based on the logical operations specified in the statement.

Step by step solution

01

Identify Variables

First, identify all the variables in the statement. In this case, the variables are \(p\), \(q\), and \(r\).
02

Create Initial Truth Table

Create a table with four columns, one for each variable \(p\), \(q\), \(r\), and one for the entire expression. There should be eight rows \(2^3 = 8\), representing all possible combinations of True (T) and False (F) for each variable.
03

Calculate \(p \wedge q\)

Calculate the logical conjunction of \(p\) and \(q\) (represented by \(p \wedge q\)) for every row and add it in a new column.
04

Calculate \(\sim(p \wedge q)\) and \(\sim r\)

Negate (\(\sim\)) the results of \(p \wedge q\) and \(r\), and add these in two new columns.
05

Calculate the Entire Expression

Now calculate the entire expression \(\sim(p \wedge q) \vee \sim r\) for every row. This is the disjunction (\(\vee\)) of the results from Step 4, and write these in the column for the entire expression.

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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. $$ \begin{aligned} &p \rightarrow q \\ &\underline{q \rightarrow r} \\ &\therefore r \rightarrow p \end{aligned} $$

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

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 \(C\).

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 The Graduate and Midnight Cowboy are shown, then the performance is sold out. Midnight Cowboy was shown and the performance was not sold out. \(\therefore\) The Graduate was not shown.

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 a metrorail system is not in operation, there are traffic delays. Over the past year there have been no traffic delays. \(\therefore\) Over the past year a metrorail system has been in operation.
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