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Chapter 3: Problem 67

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

Short Answer

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

Step by step solution

01

Understand the Logical Operators

Start by understanding what each logical operator means. In this case, there are two operators: Negation (\(\sim)) and Bi-conditional (\(\leftrightarrow\)), which work as follows:\n- Negation (\(\sim)): This operator inverts the truth value, meaning if \(p\) is true, \(\sim p\) will be false and vice versa.\n- Bi-conditional (\(\leftrightarrow\)): This operator evaluates to true if both statements on either side are exactly the same, meaning if they are both true or both false. If one statement is true and the other false, the bi-conditional operator returns false.
02

Substitute the given Truth Values

Now, substitute the given values of \(p\), \(q\), and \(r\) into the expression \(\sim p \leftrightarrow q\). In this case, \(p\) is false, \(q\) is true and \(r\) is false. We ignore \(r\) as it is not present in the expression. We therefore have \( \sim (false) \leftrightarrow true\).
03

Apply the Negation Operator

Now applying the negation operator, we know that the negation of false is true. Therefore, the expression becomes \( true \leftrightarrow true\)
04

Apply the Bi-conditional Operator

Finally, as the two statements on either side of the bi-conditional operator are true, the expression evaluates to true as per the definition of the bi-conditional operator.

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Most popular questions from this chapter

Write a valid argument on one of the following questions. If you can, write valid arguments on both sides. a. Should the death penalty be abolished? b. Should Roe v. Wade be overturned? c. Are online classes a good idea? d. Should marijuana be legalized? e. Should grades be abolished? f. Should same-sex marriage be legalized?

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 the standard forms of valid arguments to draw a valid conclusion from the given premises. The writers of My Mother the Car were told by the network to improve their scripts or be dropped from prime time. The writers of My Mother the Car did not improve their scripts. Therefore, ...

Write an original argument in words for the direct reasoning form.

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 watch Schindler's List and Milk, I am aware of the destructive nature of intolerance. Today I did not watch Schindler's List or I did not watch Milk. \(\therefore\) Today I am not aware of the destructive nature of intolerance.
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