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Chapter 3: Problem 38

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

Short Answer

Expert verified
The truth table for the statement \(p \vee(\sim q \wedge r)\) involves 3 variables p, q, and r and hence has 8 rows corresponding to the permutation of truth values for these variables. The resulting column (p or [not q and r]) is True for the rows where p is True or both q is False and r is True.

Step by step solution

01

Setting up the truth table

First you construct a table with four columns and eight rows. Label the first three columns as p, q, and r. These columns will represent the truth values of the variables. The fourth column will represent the truth value of the whole statement \(p \vee(\sim q \wedge r)\). Fill the rows in p, q, r columns with all possible combinations of truth values, which are True and False.
02

Applying NOT operator on q

This involves negating all the truth values in q column to generate the truth values of \(\sim q\). This is because NOT operation negates the input truth value.
03

Applying AND operator on \(\sim q\) and r

Apply the AND operation on the truth values you just obtained for \(\sim q\) and the truth values from the r column. Remember, result of AND operation is True if and only if both inputs are True.
04

Applying the OR operator on p and (\(\sim q \wedge r)\)

Now apply the OR operation on the truth values of p and the values obtained in step 3 for \(\sim q \wedge r\). The result of OR operation is True if at least one of the inputs is True. Fill in these values in the fourth column.

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Most popular questions from this chapter

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.) There must be a dam or there is flooding. This year there is flooding. \(\therefore\) This year there is no dam.

In Exercises 43-50, use the standard forms of valid arguments to draw a valid conclusion from the given premises. If a person is a chemist, then that person has a college degree. My best friend does not have a college degree. Therefore, ...

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

Determine whether each argument is valid or invalid. Some natural numbers are even, all natural numbers are whole numbers, and all whole numbers are integers. Thus, some integers are even.

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 tell you I cheated, I'm miserable. If I don't tell you I cheated, I'm miserable. \(\therefore\) I'm miserable.
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