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Chapter 3: Problem 62

Determine the truth value for each statement when \(p\) is false, \(q\) is true, and \(r\) is false. \((\sim p \vee q) \wedge(\sim r \vee p)\)

Short Answer

Expert verified
The truth value of the given logical expression is true.

Step by step solution

01

Understanding the NOT operation

Start by applying the NOT operation to the variables it directly applies to. This operation reverses the logical value. Since \(p\) is false, \(\sim p\) is true. Since \(r\) is false, \(\sim r\) is true.
02

Applying the OR operation

Apply the OR operation to the expressions derived above and \(q\) and exactly \(p\). The OR operation returns true if either or both of the two operands are true. Thus, \((\sim p \vee q)\) becomes true because \(\sim p\) is true and \(q\) is true, and \((\sim r \vee p)\) becomes true because \(\sim r\) is true and \(p\) is false.
03

Applying the AND operation

Finally, apply the AND operation to the results of the previous step. The AND operation returns true only when both operands are true. So, \((\sim p \vee q) \wedge(\sim r \vee p)\) becomes (true) \(\wedge\) (true) = true

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

Determine whether each argument is valid or invalid. No \(A\) are \(B\), some \(A\) are \(C\), and all \(C\) are \(D\). Thus, some \(D\) are \(B\)

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.

Use Euler diagrams to determine whether each argument is valid or invalid. All actors are artists. Sean Penn is an artist. Therefore, Sean Penn is an actor.

In Exercises 51-58, translate each argument into symbolic form. Then determine whether the argument is valid or invalid. If it was any of your business, I would have invited you. It is not, and so I did not.
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