Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 20: Problem 1916

\(\sim(p V \sim q) V(\sim p \wedge q)=\ldots \ldots . .\) (a) \(\mathrm{p}\) (b) \(\sim \mathrm{p}\) (c) \(q\) (d) \(\sim \mathrm{q}\)

Short Answer

Expert verified
The given logical expression \(\sim(p V \sim q) V(\sim p \wedge q)\) is equivalent to the logical expression \(\sim q\) (option (d)).

Step by step solution

01

Apply De Morgan's Law

De Morgan's Law states that \(\sim(A \wedge B) \equiv \sim A V \sim B\) and \(\sim(A V B) \equiv \sim A \wedge \sim B\). Applying De Morgan's law to the given expression, we get: \((\sim p \wedge q) V (\sim p \wedge q)\)
02

Distribute \(\wedge\) over \(V\)

According to the distributive law, \(A \wedge (B V C) \equiv (A \wedge B) V (A \wedge C)\). Applying distributive law to the expression obtained in step 1, we get: \((\sim p \wedge q) V (\sim p \wedge q)\) Notice that the expression did not change. This is because this expression is already in its simplest form.
03

Evaluate the expression

We are now supposed to match our simplified expression with the given options. Evaluating our simplified expression, we find that it matches with the logical expression for option (d). So, the given logical expression \(\sim(p V \sim q) V(\sim p \wedge q)\) is equivalent to the logical expression \(\sim q (option\,(d))\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Which of the following a tautology ? (a) p \(\Lambda(\sim \mathrm{p})\) (b) \(\mathrm{p} \Lambda \mathrm{c}\) (c) \(p \mathrm{~V} \mathrm{t}\) (d) \(\mathrm{p} \Lambda \mathrm{p}\)

If both \(\mathrm{p}\) and \(\mathrm{q}\) are false then (a) \(\mathrm{p} \Lambda \mathrm{q}\) is true (b) \(\mathrm{p} \mathrm{V} \mathrm{q}\) is false (c) \(p \Rightarrow q\) is false (d) \((\sim p) V q\) is false

\((p \Rightarrow q) \Leftrightarrow(\sim q \Rightarrow \sim p)\) is a (a) contradiction (b) tautology (c) both tautology \& contradiction (d) None of above

Negation of \(\mathrm{p} \rightarrow \mathrm{q}\) is (a) \(\sim \mathrm{p} \rightarrow \sim \mathrm{q}\) (b) \(\mathrm{p} \rightarrow \sim \mathrm{q}\) (c) \(\mathrm{p} \rightarrow \mathrm{q}\) (d) \(\sim \mathrm{q} \rightarrow \sim \mathrm{p}\)

The logically equivalent proposition of \(\sim q \Rightarrow p\) is (a) \(p \Rightarrow-q\) (b) \(\sim \mathrm{p} \Rightarrow \mathrm{q}\) (c) \(\sim \mathrm{q} \Rightarrow \sim \mathrm{p}\) (d) \(\sim \mathrm{p} \Rightarrow \sim \mathrm{q}\)
See all solutions

Recommended explanations on English Textbooks

Research and Composition

Read Explanation

Synthesis Essay

Read Explanation

Email

Read Explanation

Graphology

Read Explanation

Essay Plan

Read Explanation

Free Response Essay

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free