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Chapter 1: Problem 36

Show that \((p \wedge q) \rightarrow r\) and \((p \rightarrow r) \wedge(q \rightarrow r)\) are not logically equivalent.

Short Answer

Expert verified
They are not logically equivalent because their truth values differ in some cases.

Step by step solution

01

Understand Logical Equivalence

Two statements are logically equivalent if they have the same truth value in every possible scenario. To determine whether \((p \wedge q) \rightarrow r\) and \((p \rightarrow r) \wedge(q \rightarrow r)\) are logically equivalent, a truth table can be used.
02

Set Up the Truth Table

Create a truth table that includes all possible truth values for propositions \(p\), \(q\), and \(r\).
03

Fill in Columns for Basic Propositions

List all possible combinations of truth values for \(p\), \(q\), and \(r\). The table will have 8 rows for the 8 possible combinations.
04

Calculate \((p \wedge q) \rightarrow r\)

Determine the truth value of \((p \wedge q)\), then use this to find the truth value of \((p \wedge q) \rightarrow r\) for each row in the truth table.
05

Calculate \(p \rightarrow r\) and \(q \rightarrow r\)

For each row, determine the truth values of \(p \rightarrow r\) and \(q \rightarrow r\).
06

Calculate \((p \rightarrow r) \wedge (q \rightarrow r)\)

Using the results from Step 5, determine the truth value of \((p \rightarrow r) \wedge (q \rightarrow r)\) for each row in the truth table.
07

Compare the Columns

Compare the truth values obtained for \((p \wedge q) \rightarrow r\) and \((p \rightarrow r) \wedge (q \rightarrow r)\) for each row. If there is at least one row where the truth values differ, the statements are not logically equivalent.
08

Conclusion

Since there are rows in the truth table where \((p \wedge q) \rightarrow r\) and \((p \rightarrow r) \wedge (q \rightarrow r)\) have different truth values, they are not logically equivalent.

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Key Concepts

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

Truth Table
A truth table is a mathematical table used to determine if a compound statement is true or false based on the truth values of its components. Each row represents a possible combination of truth values for the variables involved. In our exercise, we create a truth table to compare \((p \wedge q) \rightarrow r\) and \((p \rightarrow r) \wedge(q \rightarrow r)\). First, list all possible truth values for p, q, and r. There are 8 combinations (2^3, since each variable can be either true or false). Have columns for p, q, r, \((p \wedge q)\), \((p \wedge q) \rightarrow r\), \((p \rightarrow r)\), and \((q \rightarrow r)\). Then, calculate the values for each column using logical rules.
Implication in Logic
Implication in logic, denoted by \( \rightarrow \), is a logical connective that represents 'if...then...' statements. The implication \( p \rightarrow r \) is true unless p is true and r is false. In our exercise, we look at two implications: \((p \wedge q) \rightarrow r\) and \((p \rightarrow r) \wedge (q \rightarrow r)\). For \((p \wedge q) \rightarrow r\), it's true if r is true or \((p \wedge q)\) is false. For \((p \rightarrow r) \wedge (q \rightarrow r)\), both implications must be true.
Logical Equivalence
Logical equivalence means two statements have the same truth value in every possible situation. To prove logical equivalence (or not), we can use truth tables to compare the statements. Statements \((p \wedge q) \rightarrow r\) and \((p \rightarrow r) \wedge (q \rightarrow r)\) are not logically equivalent if there is at least one row in the truth table where they have different values. In our example, comparing the truth values from our truth table shows they differ, proving they are not logically equivalent.
Conjunction and Disjunction in Logic
Conjunction and disjunction are basic logical operations. Conjunction (\( p \wedge q \)) means 'p and q' must both be true for the statement to be true. Disjunction (\( p \vee q \)) means 'p or q' (or both) must be true for the statement to be true. In our problem, we use conjunction (\( p \wedge q \)) within the implication \((p \wedge q) \rightarrow r\) to determine its truth value. Understanding these operations helps us build and decipher complex logical statements used in the truth table.

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Most popular questions from this chapter

Suppose that \(a\) and \(b\) are odd integers with \(a \neq b .\) Show there is a unique integer \(c\) such that \(|a-c|=|b-c|\)

Use rules of inference to show that if \(\forall x(P(x) \vee Q(x))\) \(\forall x(\neg Q(x) \vee S(x)), \quad \forall x(R(x) \rightarrow \neg S(x)),\) and \(\exists x \neg P(x)\) are true, then \(\exists x \neg R(x)\) is true.

Express each of these statements using quantifiers. Then form the negation of the statement, so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase "It is not the case that.") a) Some old dogs can learn new tricks. b) No rabbit knows calculus. c) Every bird can fly. d) There is no dog that can talk. e) There is no one in this class who knows French and Russian.

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