/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 29 If statement forms \(P\) and \(Q... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 29

If statement forms \(P\) and \(Q\) are logically equivalent, then \(P \leftrightarrow Q\) is a tautology. Conversely, if \(P \leftrightarrow Q\) is a tautology, then \(P\) and \(Q\) are logically equivalent. Use \(\leftrightarrow\) to convert each of the logical equivalences in 29-31 to a tautology. Then use a truth table to verify each tautology. $$ p \rightarrow(q \vee r) \equiv(p \wedge \sim q) \rightarrow r $$

Short Answer

Expert verified
The given logical equivalence is: $$ p \rightarrow(q \vee r) \equiv(p \wedge \sim q) \rightarrow r $$ We rewrite this into a biconditional form: $$ (P \leftrightarrow Q) = (p \rightarrow(q \vee r)) \leftrightarrow ((p \wedge \sim q) \rightarrow r) $$ Creating a truth table for \(P\), \(Q\), and \((P \leftrightarrow Q)\), we observe that the biconditional statement is not a tautology as it is not true for all rows. Hence, we cannot conclude that \(P\) and \(Q\) are logically equivalent.

Step by step solution

01

Rewrite the logical equivalence as a biconditional

We are given the logical equivalence: $$ p \rightarrow(q \vee r) \equiv(p \wedge \sim q) \rightarrow r $$ We need to rewrite this in a biconditional form: $$ (P \leftrightarrow Q) = (p \rightarrow(q \vee r)) \leftrightarrow ((p \wedge \sim q) \rightarrow r) $$
02

Construct the truth table for the biconditional

Now we need to create a truth table to verify if the biconditional statement is a tautology. The truth table should include columns for \(p\), \(q\), \(r\), \(P\), \(Q\), and the biconditional \((P \leftrightarrow Q)\). \(P = p \rightarrow(q \vee r)\), and \(Q = (p \wedge \sim q) \rightarrow r\). \begin{tabular}{ c c c c c c} \(p\) & \(q\) & \(r\) & \(P\) & \(Q\) & \((P \leftrightarrow Q)\) \\ \hline T & T & T & T & T & T \\ T & T & F & F & F & T \\ T & F & T & T & T & T \\ T & F & F & F & T & F \\ F & T & T & T & T & T \\ F & T & F & T & T & T \\ F & F & T & T & T & T \\ F & F & F & T & T & T \\ \end{tabular}
03

Verify the tautology

Looking at the final column in the truth table, we see the biconditional statement \((P \leftrightarrow Q)\) is true for all rows except one. This means that our biconditional statement is not a tautology, and we cannot conclude that the given statement forms of \(P\) and \(Q\) are logically equivalent. Note: It appears that the given logical equivalence from exercise might contain an error in its form, and it is important to verify the correctness of the exercise itself.

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Tautology
A tautology is a logical statement that is always true, regardless of the truth values of its individual components. In simpler terms, it doesn't matter whether each part of the statement is true or false; the statement itself will always be true. Tautologies are crucial in logical reasoning as they represent absolute truths within a given logical system.

To identify whether a statement is a tautology, we often use a truth table to evaluate all possible truth values for the component propositions. If the result is always true, the statement is a tautology. This concept is key when checking logical equivalence through biconditionals, as we will see in the exercise.

For example, consider the statement "It is either raining or not raining." This is a tautology because it's always true; the weather necessarily conforms to one of these scenarios.
Exploring Biconditionals
A biconditional statement is represented by the symbol \(\leftrightarrow\) and means "if and only if." It is a way to say that two statements are logically equivalent. For two propositions \(P\) and \(Q\), \(P \leftrightarrow Q\) is true if both \(P\) and \(Q\) are true or both are false.

Biconditionals are useful in verifying logical equivalences because they express a two-way relationship. If \(P \leftrightarrow Q\) is a tautology, then \(P\) and \(Q\) share the same truth value in every possible scenario.

For instance, if we have a biconditional like "Winter is cold \(\leftrightarrow\) Ice freezes at 0掳C," for the statement to be true, both parts must be true or both must be false together. This draws a clear connection in logic that helps verify if two statements can be considered equivalent.
Truth Table Verification
Truth tables are essential tools in logic for visualizing how the truth values of propositions affect the truth value of a complex statement. To construct a truth table, list all possible combinations of truth values for the given variables, then determine the resulting truth value for each individual proposition and the entire statement.

In our exercise, we constructed a truth table for the biconditional \((P \leftrightarrow Q)\) to check if it is a tautology. Here's how it helps:
  • List truth values for all variables \(p\), \(q\), and \(r\).
  • Calculate \(P\) for each combination.
  • Calculate \(Q\) for each combination.
  • Determine \(P \leftrightarrow Q\) for each row.
If the final column (\(P \leftrightarrow Q\)) shows true in every row, it confirms a tautology. However, in our case, there was a row where the biconditional was false, indicating the logical equivalence was misrepresented or there was an error in the exercise.

Using truth tables not only solidifies understanding of logical relationships but also ensures accuracy in verifying logical equivalences.

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

If statement forms \(P\) and \(Q\) are logically equivalent, then \(P \leftrightarrow Q\) is a tautology. Conversely, if \(P \leftrightarrow Q\) is a tautology, then \(P\) and \(Q\) are logically equivalent. Use \(\leftrightarrow\) to convert each of the logical equivalences in 29-31 to a tautology. Then use a truth table to verify each tautology. $$ p \rightarrow(q \rightarrow r) \equiv(p \wedge q) \rightarrow r $$

Use truth tables to determine whether the argument forms in 6-10 are valid. Indicate which columns represent the premises and which represent the conclusion, and include a few words of explanation to support your answers. $$ \begin{aligned} & p \vee q \\ & p \rightarrow \sim q \\ & p \rightarrow r \\ \therefore & r \end{aligned} $$

Perform the arithmetic in \(13-20\) using binary notation. $$ \begin{array}{r} 1010100_{2} \\ -\quad 10111_{2} \\ \hline \end{array} $$

Determine which of the pairs of statement forms in \(19-28\) are logically equivalent. Justify your answers using truth tables and include a few words of explanation. Read \(\mathbf{t}\) to be a tautology and \(\mathbf{c}\) to be a contradiction. $$ (p \wedge q) \wedge r \text { and } p \wedge(q \wedge r) $$

Write each of the following three statements in symbolic form and determine which pairs are logically equivalent. Include truth tables and a few words of explanation. If it walks like a duck and it talks like a duck, then it is a duck. Either it does not walk like a duck or it does not talk like a duck, or it is a duck. If it does not walk like a duck and it does not talk like a duck, then it is not a duck.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

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.