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Chapter 3: Problem 57

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

Short Answer

Expert verified
The truth value of the statement \(\sim(p \wedge q) \vee r\) is true when \(p\) is false, \(q\) is true, and \(r\) is false.

Step by step solution

01

Evaluate the Expression inside the Brackets

First let鈥檚 handle \(p \wedge q\), which is 'AND' operation. 'AND' is true only if both its operands are true. So, with \(p\) being false and \(q\) being true, the result of \(p \wedge q\) is false.
02

Evaluate the Negation

Now apply the negation operator to the result from step 1. Here, the 'NOT' of false i.e., \(\sim(false)\) is true.
03

Evaluate the Final Expression

Finally, solve the 'OR' operation using the result from Step 2 (which is true) and \(r\) (which is false). 'OR' is true if either of its operands are true. So, in our case, true \(\vee\) false gives us true.

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Most popular questions from this chapter

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

Use a truth table to determine whether the symbolic form of the argument is valid or invalid. $$ \begin{aligned} &p \rightarrow q \\ &\underline{q \rightarrow r} \\ &\therefore \sim p \rightarrow \sim r \end{aligned} $$

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 am at the beach, then I swim in the ocean. If I swim in the ocean, then I feel refreshed. \(\therefore\) If I am at the beach, then I feel refreshed.

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 a metrorail system is not in operation, there are traffic delays. Over the past year there have been no traffic delays. \(\therefore\) Over the past year a metrorail system has been in operation.

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.
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