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Chapter 3: Problem 38

Use a truth table to determine whether each statement is a tautology, a self- contradiction, or neither. \([(p \vee q) \wedge \sim q] \rightarrow p\)

Short Answer

Expert verified
By constructing a truth table and evaluating the given statement for all truth value combinations, one can determine whether it is a tautology, contradiction, or neither.

Step by step solution

01

Constructing the Truth Table

First, create a truth table that covers all possible truth value combinations for \(p\) and \(q\). Hence, there would be four rows: \(T, T\), \(T, F\), \(F, T\), and \(F, F\)
02

Computing Inner Expression

Next, calculate the value of the expression within the brackets using the truth values of \(p\) and \(q\). This expression is \((p \vee q) \wedge \sim q\) and involves an 'or' operation, 'and' operation and a 'not' operation.
03

Obtaining Final Result

The final step is to evaluate the entire statement, \([(p \vee q) \wedge \sim q] \rightarrow p\). Use the values obtained in Step 2, along with the truth values of \(p\), to find out whether the statement is true or false in each case.

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

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

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 all people obey the law, then no jails are needed. Some people do not obey the law. \(\therefore\) Some jails are needed.

Use Euler diagrams to determine whether each argument is valid or invalid. All insects have six legs. No spiders are insects. Therefore, no spiders have six legs.

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 not reading. \(\therefore\) It is neither raining nor snowing.
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