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Chapter 3: Problem 29

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

Short Answer

Expert verified
The truth table for the given logical statement \((p \wedge r) \leftrightarrow \sim(q \vee r)\) is achieved by listing all possible combinations of true and false values for the variables p, q, r, then solving for each component of the statement and finally the entire statement for each combinations. Equivalence returns true only when both sides have same values.

Step by step solution

01

Identify the variables

Identify the variables used in the statement. In this case, the variables are p, q, and r.
02

List all possible variable combinations

As we have three variables (p, q, r) we have \(2^3 = 8\) different possible combinations of truth values.
03

Break down the main expression

The main expression \((p \wedge r) \leftrightarrow \sim(q \vee r)\) consists of sub-expressions. Break it down into smaller, more manageable bits. In particular, focus on evaluating \(p \wedge r\), \(q \vee r\), and \sim (q \vee r) separately.
04

Calculate the expressions

Calculate the expressions for each combination of variable values. Note that AND (\wedge) only yields true if both operands are true, OR (\vee) yields true if at least one of the operands is true, and negation (\sim) inverts the boolean value of its operand.
05

Combine the results

Use the results of \(p \wedge r\) and \sim(q \vee r) to calculate the value of the whole expression \((p \wedge r) \leftrightarrow \sim(q \vee r)\) for each combination of variable values. Remember the equivalence (\leftrightarrow) yields true if the values on both sides are identical, i.e., both true or both false.

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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 I'm tired, I'm edgy. If I'm edgy, I'm nasty. \(\therefore\) If I'm tired, I'm nasty.

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

Use the standard forms of valid arguments to draw a valid conclusion from the given premises. If I vacation in Paris, I eat French pastries. If I eat French pastries, I gain weight. Therefore, ...

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 tell you I cheated, I'm miserable. If I don't tell you I cheated, I'm miserable. \(\therefore\) I'm miserable.

Draw a valid conclusion from the given premises. Then use a truth table to verify your answer. If you only spoke when spoken to and I only spoke when spoken to, then nobody would ever say anything. Some people do say things. Therefore, ...
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