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Chapter 3: Problem 13

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

Short Answer

Expert verified
The truth table will have 8 rows, with the overall value of the statement depending on the specific true or false values of \(p\), \(q\) and \(r\).

Step by step solution

01

Identify the Logical Symbols

Before constructing the truth table, it's necessary to identify each symbol in the logical statement. \(\sim\) stands for NOT, \(\wedge\) stands for AND, and \(\rightarrow\) stands for IF-THEN.
02

Construct the Initial Truth Table

Next, create a table with four columns, one for each of the variables \(p\), \(q\) and \(r\) and one for the complex statement \(\sim r \wedge(\sim q \rightarrow p)\). Each of the variable columns will have 2n rows (where n is the number of variables), filled with alternating T's (True) and F's (False). Here you should start with short alternations and then double the length for each variable.
03

Determine the Truth Values of the Compound Statements

Based on the truth values of \(p\), \(q\) and \(r\), define the overall values of the compound statement. Remember that for \(\sim r\), the value is true when \(r\) is false and vice versa. The \(\rightarrow\) operator has the value true except in the case when its first operand is true and the second is false.
04

Compute the Final Truth Table

Now, it's time to find out the truth value of the whole complex statement \(\sim r \wedge(\sim q \rightarrow p)\). It will be true only if both \(\sim r\) is true and \((\sim q \rightarrow p)\) are true.

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Most popular questions from this chapter

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 \(6 .\) Therefore, 8 is not a multiple of 3 .

Write an original argument in words that has a true conclusion, yet is invalid.

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 rains or snows, then I read. I am reading. \(\therefore\) It is raining or snowing.

In the Sixth Meditation, Descartes writes I first take notice here that there is a great difference between the mind and the body, in that the body, from its nature, is always divisible and the mind is completely indivisible. Descartes's argument can be expressed as follows: All bodies are divisible. No minds are divisible. Therefore, no minds are bodies. Use an Euler diagram to determine whether the argument is valid or invalid.

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.
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