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Chapter 3: Problem 31

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

Short Answer

Expert verified
The truth table for the given logical statement is: TT -> F; TF -> T; FT -> F; FF -> T. This denotes that the overall expression is false when both p and q are true, and both are false; and true when p is true and q is false, and vice versa.

Step by step solution

01

List all possible combinations of truth values for p and q

We should start by writing down all the possible combinations of truth values for the variables p and q, which are two in this case. This gives us a total of four combinations: TT (both true), TF (true and false), FT (false and true), FF (both false).
02

Apply the negation operator

Next, apply the negation operator to p and q. This will invert the truth value of the given variable. Compute \( \sim p \) and \( \sim q \) for each combination.
03

Perform the conjunction operation

Compute the conjunction operation \(p \wedge \sim q\) for the given truth value combination. Remember, the result is true (T) if both operands are true, and false (F) otherwise.
04

Compute the final disjunction operation

Third, for each row in our truth table, compute the disjunction \( \sim p \vee (p \wedge \sim q) \). A disjunction operation is true if at least one of its operands is true, otherwise, it is false. This gives us the desired truth value.

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Most popular questions from this chapter

Describe what is meant by a valid argument.

Use Euler diagrams to determine whether each argument is valid or invalid. All multiples of 6 are multiples of 3 . Eight is not a multiple of \(6 .\) Therefore, 8 is not a multiple of 3 .

Use Euler diagrams to determine whether each argument is valid or invalid. All dogs have fleas. Some dogs have rabies. Therefore, all dogs with rabies have fleas.

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.

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. "I do know that this pencil exists; but I could not know this if Hume's principles were true. Therefore, Hume's principles, one or both of them, are false."
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