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6. Construct a truth table for the following statement. (p V ~q) → 9

6. Construct a truth table for the following statement. (p V ~q) → 9

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Certainly! Below is a transcription suitable for an educational website: --- **Exercise 6.** Construct a truth table for the following statement: \((p \lor \sim q) \rightarrow q\) --- To solve this, start by considering the logical operators involved: 1. **Disjunction (\(\lor\))**: This operator is true if at least one of the operands is true. 2. **Negation (\(\sim\))**: This operator inverts the truth value of its operand. 3. **Implication (\(\rightarrow\))**: This operator is false only when the first operand is true and the second operand is false. The truth table will examine all possible truth values of \(p\) and \(q\): | \(p\) | \(q\) | \(\sim q\) | \(p \lor \sim q\) | \((p \lor \sim q) \rightarrow q\) | |-------|-----|------|-----------------|-------------------------| | T | T | F | T | T | | T | F | T | T | F | | F | T | F | F | T | | F | F | T | T | F | Explanation of each column: - **\(p\) and \(q\)**: Initial truth values we are considering. - **\(\sim q\)**: The negation of \(q\). - **\(p \lor \sim q\)**: The result of the disjunction between \(p\) and \(\sim q\). - **\((p \lor \sim q) \rightarrow q\)**: The final implication we want to evaluate. This table effectively illustrates both the intermediate logical outcomes and the final conclusion of the statement for all possible logical inputs.
Transcribed Image Text:Certainly! Below is a transcription suitable for an educational website: --- **Exercise 6.** Construct a truth table for the following statement: \((p \lor \sim q) \rightarrow q\) --- To solve this, start by considering the logical operators involved: 1. **Disjunction (\(\lor\))**: This operator is true if at least one of the operands is true. 2. **Negation (\(\sim\))**: This operator inverts the truth value of its operand. 3. **Implication (\(\rightarrow\))**: This operator is false only when the first operand is true and the second operand is false. The truth table will examine all possible truth values of \(p\) and \(q\): | \(p\) | \(q\) | \(\sim q\) | \(p \lor \sim q\) | \((p \lor \sim q) \rightarrow q\) | |-------|-----|------|-----------------|-------------------------| | T | T | F | T | T | | T | F | T | T | F | | F | T | F | F | T | | F | F | T | T | F | Explanation of each column: - **\(p\) and \(q\)**: Initial truth values we are considering. - **\(\sim q\)**: The negation of \(q\). - **\(p \lor \sim q\)**: The result of the disjunction between \(p\) and \(\sim q\). - **\((p \lor \sim q) \rightarrow q\)**: The final implication we want to evaluate. This table effectively illustrates both the intermediate logical outcomes and the final conclusion of the statement for all possible logical inputs.
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