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Chapter 3: Problem 40

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

Short Answer

Expert verified
The truth table for the compound statement \((r \vee \sim p) \wedge \sim q\) will be a table with eight rows (representing all possible combinations of truth values for \(p\), \(q\) and \(r\)) and five columns (for \(p\), \(q\), \(r\), \(r \vee \sim p\) and \((r \vee \sim p) \wedge \sim q\)). In this table, each row will provide the truth values of the compound statement based on the specific combination of the values of \(p\), \(q\) and \(r\).

Step by step solution

01

Define the Variables

Define the three variables included in the compound statement: \(p\), \(q\) and \(r\). These are the basic elements of our formula and can each take the binary values True (denoted as T) or False (denoted as F).
02

List all possible combinations of truth values

We need to specify every possible combination of truth values for these three variables, which will be eight in total (2^3). This results in a standard truth table structure where each row represents a different combination of truth values for \(p\), \(q\) and \(r\).
03

Compute the value of \(r \vee \sim p\) for each row

Calculate the output of the operation \(r \vee \sim p\). Here, \(\sim p\) is the negation of \(p\) and \(r \vee \sim p\) gives the output as true if either \(r\) is true or \(p\) is false.
04

Compute \(\sim q\) for each row

The 'NOT' operation alters the truth value of its operand. So compute \(\sim q\) which gives the output true if \(q\) initially was false.
05

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

Finally, compute the result of \((r \vee \sim p) \wedge \sim q\). The 'AND' operation will only be true if both the operands are true. Therefore, we compare the results of step 3 and step 4 in an 'AND' operation.

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Most popular questions from this chapter

If you are given an argument in words that contains two premises and a conclusion, describe how to determine if the argument is valid or invalid.

Use Euler diagrams to determine whether each argument is valid or invalid. All multiples of 6 are multiples of 3 . Eight is not a multiple of 3 . Therefore, 8 is not a multiple of 6 .

This is an excerpt from a 1967 speech in the U.S. House of Representatives by Representative Adam Clayton Powell: He who is without sin should cast the first stone. There is no one here who does not have a skeleton in his closet. I know, and I know them by name. Powell's argument can be expressed as follows: No sinner is one who should cast the first stone. All people here are sinners. Therefore, no person here is one who should cast the first stone. Use an Euler diagram to determine whether the argument is valid or invalid.

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. It is the case that \(x<5\) or \(x>8\), but \(x \geq 5\), so \(x>8\).

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 it is hot and humid, I complain. It is not hot or it is not humid. \(\therefore\) I am not complaining.
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