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Chapter 3: Problem 26

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

Short Answer

Expert verified
The truth table for the statement \( \sim q \wedge p \) would look like:\[\begin{array}{|c|c|c|}\hlinep & q & \sim q \wedge p \\hline1 & 1 & 0 \1 & 0 & 1 \0 & 1 & 0 \0 & 0 & 0 \\hline\end{array}\]

Step by step solution

01

Identify the Variables

The variables in the logical statement are p and q. These are the variables that we will input into our truth table. Normally, a binary variable can take the value of either true (often represented as 1) or false (often represented as 0), giving us four possible combinations of p and q: (1,1), (1,0), (0,1), and (0,0).
02

Create the Initial Part of the Table

Next, create a table with three columns. The first two columns represent the variables p and q. Their true (1) or false (0) values will be written in the rows under these columns. The third column represents the entire logical statement \( \sim q \wedge p \). The outcomes of this logical statement corresponding to each combination of p and q will be written in this column.
03

Populate the Values

After setting up the table, insert the truth values of each variable combination. Note that the \(\sim\) operator changes the truth value of the proposition it is attached to. Since \(\sim\) is attached to q, use the opposite of each truth value that you wrote for q. Since the \(\wedge\) operator returns true only if both variables it affects are true, if both \( \sim q \) and p are true, write 1 under the \( \sim q \wedge p \) column. Else, write 0.

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Most popular questions from this chapter

Use Euler diagrams to determine whether each argument is valid or invalid. All comedians are funny people. Some funny people are professors. Therefore, some comedians are professors.

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

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I made Euler diagrams for the premises of an argument and one of my possible diagr芒ms did not illustraate the conclusion, so the argument is invalid.

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

Use a truth table to determine whether the symbolic form of the argument is valid or invalid. \(\sim p \vee q\) P ____ \(\therefore q\)
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