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Determine the validity of each the following arguments. If the argument is one of those listed in the text, name it. She uses e-commerce or she pays by credit card. She does not pay by credit card. She uses e-commerce.

Determine the validity of each the following arguments. If the argument is one of those listed in the text, name it. She uses e-commerce or she pays by credit card. She does not pay by credit card. She uses e-commerce.

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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### Determining the Validity of Arguments **Exercise:** Determine the validity of each of the following arguments. If the argument is one of those listed in the text, name it. 1. She uses e-commerce or she pays by credit card. 2. She does not pay by credit card. 3. ____________________________ 4. She uses e-commerce. This exercise asks you to assess if the given arguments lead to a valid conclusion. **Given Argument Breakdown:** - **Premise 1:** She uses e-commerce or she pays by credit card. - **Premise 2:** She does not pay by credit card. - **Conclusion:** She uses e-commerce. ### Explanation of the Argument In logical terms, this argument employs the concept of disjunction (logical "or") and negation to arrive at a conclusion. - **Symbolic Form:** - P1: \( P \lor Q \) (She uses e-commerce or she pays by credit card.) - P2: \( \neg Q \) (She does not pay by credit card.) - Conclusion: \( P \) (She uses e-commerce.) According to the rules of logical reasoning, specifically Disjunctive Syllogism, if a disjunction (\( P \lor Q \)) is true and one of the components (Q) is false (\( \neg Q \)), then the other component (P) must be true. ### Conclusion From the given premises, we can validly conclude that she uses e-commerce. This argument is an example of a Disjunctive Syllogism. **Note:** Understanding and identifying the form of logical arguments is a critical skill in logical reasoning and helps in constructing valid conclusions from given premises.
Transcribed Image Text:### Determining the Validity of Arguments **Exercise:** Determine the validity of each of the following arguments. If the argument is one of those listed in the text, name it. 1. She uses e-commerce or she pays by credit card. 2. She does not pay by credit card. 3. ____________________________ 4. She uses e-commerce. This exercise asks you to assess if the given arguments lead to a valid conclusion. **Given Argument Breakdown:** - **Premise 1:** She uses e-commerce or she pays by credit card. - **Premise 2:** She does not pay by credit card. - **Conclusion:** She uses e-commerce. ### Explanation of the Argument In logical terms, this argument employs the concept of disjunction (logical "or") and negation to arrive at a conclusion. - **Symbolic Form:** - P1: \( P \lor Q \) (She uses e-commerce or she pays by credit card.) - P2: \( \neg Q \) (She does not pay by credit card.) - Conclusion: \( P \) (She uses e-commerce.) According to the rules of logical reasoning, specifically Disjunctive Syllogism, if a disjunction (\( P \lor Q \)) is true and one of the components (Q) is false (\( \neg Q \)), then the other component (P) must be true. ### Conclusion From the given premises, we can validly conclude that she uses e-commerce. This argument is an example of a Disjunctive Syllogism. **Note:** Understanding and identifying the form of logical arguments is a critical skill in logical reasoning and helps in constructing valid conclusions from given premises.
**Title: Verification of Logical Argument Validity** **Introduction:** In this tutorial, we will verify the validity of a logical argument using propositional logic techniques. **Given Argument:** The argument to verify consists of the following premises and conclusion: 1. \( p \rightarrow q \) 2. \( q \rightarrow r \) 3. Conclusion: \( (p \lor q) \rightarrow r \) **Steps for Verification:** 1. **Understand the Premises and Conclusion:** - Premise 1: \( p \rightarrow q \) - This indicates that if proposition \( p \) is true, then proposition \( q \) must also be true. - Premise 2: \( q \rightarrow r \) - This indicates that if proposition \( q \) is true, then proposition \( r \) must also be true. - Conclusion: \( (p \lor q) \rightarrow r \) - This indicates that if either proposition \( p \) or \( q \) (or both) are true, then proposition \( r \) must be true. 2. **Construct a Truth Table:** To verify the validity, we construct a truth table considering the truth values for each proposition \( p \), \( q \), and \( r \). | \( p \) | \( q \) | \( r \) | \( p \rightarrow q \) | \( q \rightarrow r \) | \( p \lor q \) | \( (p \lor q) \rightarrow r \) | |--------|--------|--------|----------------------|----------------------|----------------|--------------------------| | T | T | T | T | T | T | T | | T | T | F | T | F | T | F | | T | F | T | F | T | T | T | | T | F | F | F | T | T | F | | F | T | T | T | T | T | T | | F | T
Transcribed Image Text:**Title: Verification of Logical Argument Validity** **Introduction:** In this tutorial, we will verify the validity of a logical argument using propositional logic techniques. **Given Argument:** The argument to verify consists of the following premises and conclusion: 1. \( p \rightarrow q \) 2. \( q \rightarrow r \) 3. Conclusion: \( (p \lor q) \rightarrow r \) **Steps for Verification:** 1. **Understand the Premises and Conclusion:** - Premise 1: \( p \rightarrow q \) - This indicates that if proposition \( p \) is true, then proposition \( q \) must also be true. - Premise 2: \( q \rightarrow r \) - This indicates that if proposition \( q \) is true, then proposition \( r \) must also be true. - Conclusion: \( (p \lor q) \rightarrow r \) - This indicates that if either proposition \( p \) or \( q \) (or both) are true, then proposition \( r \) must be true. 2. **Construct a Truth Table:** To verify the validity, we construct a truth table considering the truth values for each proposition \( p \), \( q \), and \( r \). | \( p \) | \( q \) | \( r \) | \( p \rightarrow q \) | \( q \rightarrow r \) | \( p \lor q \) | \( (p \lor q) \rightarrow r \) | |--------|--------|--------|----------------------|----------------------|----------------|--------------------------| | T | T | T | T | T | T | T | | T | T | F | T | F | T | F | | T | F | T | F | T | T | T | | T | F | F | F | T | T | F | | F | T | T | T | T | T | T | | F | T
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