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Chapter 3: Problem 69

Write each statement in symbolic form and construct a truth table. Then indicate under what conditions, if any, the compound statement is true. It is not true that \(x \leq 3\) or \(x \geq 7\), but \(x>3\) and \(x<7\).

Short Answer

Expert verified
The symbolic form of the given compound statement is \(\neg(p \lor q) \land (r \land s)\), a translation of 'It is not true that \(x \leq 3\) or \(x \geq 7\), but \(x > 3\) and \(x < 7\)'. The conditions under which this statement is true can be found by constructing a truth table.

Step by step solution

01

Identify the Simple Statements

First, we break down the given statement into simpler ones: \(p: x \leq 3\), \(q: x \geq 7\), \(r: x > 3\), \(s: x < 7\).
02

Translate Compound Statement

The compound statement 'It is not true that \(p\) or \(q\), but \(r\) and \(s\)' translates to \(\neg (p \lor q) \land (r \land s)\).
03

Construct a Truth Table

We construct a truth table with columns for \(p\), \(q\), \(r\), \(s\), \(p \lor q\), \(\neg (p \lor q)\), \(r \land s\), and the final compound statement \(\neg (p \lor q) \land (r \land s)\). For each row, we evaluate these statements assuming all possible combinations of truth values of \(p\), \(q\), \(r\), and \(s\)
04

Identify Conditions

The final step is to identify the conditions under which the compound statement is true. This is done by looking at the final column of the truth table in the previous step.

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