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Chapter 3: Problem 25

Use De Morgan's laws to write a statement that is equivalent to the given statement. \(p \rightarrow(q \vee \sim r)\)

Short Answer

Expert verified
The equivalent statement using De Morgan's laws is \(\sim(\sim p \wedge (\sim q \wedge r))\).

Step by step solution

01

Convert Implication to Disjunction

An implication is equivalent to a disjunction. The implication \(p \rightarrow (q \vee \sim r)\) can be rewritten in disjunctive form: \(\sim p \vee (q \vee \sim r)\)
02

Apply De Morgan's law

Now, apply De Morgan's law. When switching the disjunction \(\sim p \vee (q \vee \sim r)\) to its equivalent conjunction, we get \(\sim(\sim p \wedge \sim(q \vee \sim r))\).
03

Simplify the expression

Now we need to simplify the expression inside the conjunction. Apply De Morgan's law again within the brackets to get: \(\sim(\sim p \wedge (\sim q \wedge r))\).

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Most popular questions from this chapter

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 we are to have peace, we must not encourage the competitive spirit. If we are to make progress, we must encourage the competitive spirit. \(\therefore\) We do not have peace and we do not make progress.

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