### Determining the Validity of a Symbolic Argument Using a Truth TableIn logic, a crucial skill is determining whether an argument is valid or invalid. One way to do this is by using a truth table. Examine the following argument and determine its validity:#### Argument:1. $ p \rightarrow q $2. $ q \rightarrow p $3. $ \therefore p \vee q $#### Question:Choose the correct answer below:- O The argument is invalid.- O The argument is valid.Click to select your answer.### Explanation:1. **Symbols Explanation**: - $ p \rightarrow q $: If $ p $, then $ q $. - $ q \rightarrow p $: If $ q $, then $ p $. - $ \therefore p \vee q $: Therefore, $ p $ or $ q $.2. **Building a Truth Table**: To determine the validity, let's create a truth table for these statements. A truth table lists all possible truth values for the propositions $ p $ and $ q $ and determines the truth values of the compound statements. | $ p $ | $ q $ | $ p \rightarrow q $ | $ q \rightarrow p $ | $ p \vee q $ | Both premises true? | Conclusion true if premises true? | |:------:|:------:|:--------------------:|:--------------------:|:-------------:|:------------------:|:------------------------------:| | T | T | T | T | T | True | True | | T | F | F | T | T | False | True | | F | T | T | F | T | False | True | | F | F | T | T | F | True | False |3. **Analysis**: - When both $ p $ and $ q $ are true (first row), $ p \rightarrow q $ is true, $ q \rightarrow p $ is true, and $ p \vee q $ is true. Both the

p -> q means " if p then q" q -> p means " if q then p"

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Use a truth table to determine whether the symbolic form of the argument is valid or invalid

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Logic and Proofs

Discrete mathematics deals with the separated values of data such as integers. It is widely used in the practical field of computer science and mathematics. By knowing discrete mathematics one can improve their reasoning and problem-solving capabilities. Discrete mathematics became popular after the boom of computers which operate in discrete steps and store data in discrete bits. It also useful in studying computer programming, algorithms, and cryptography.

Question

### Determining the Validity of a Symbolic Argument Using a Truth Table

In logic, a crucial skill is determining whether an argument is valid or invalid. One way to do this is by using a truth table. Examine the following argument and determine its validity:

#### Argument:
1. $ p \rightarrow q $
2. $ q \rightarrow p $
3. $ \therefore p \vee q $

#### Question:
Choose the correct answer below:

- O The argument is invalid.
- O The argument is valid.

Click to select your answer.

### Explanation:

1. **Symbols Explanation**:
- $ p \rightarrow q $: If $ p $, then $ q $.
- $ q \rightarrow p $: If $ q $, then $ p $.
- $ \therefore p \vee q $: Therefore, $ p $ or $ q $.

2. **Building a Truth Table**:
To determine the validity, let's create a truth table for these statements. A truth table lists all possible truth values for the propositions $ p $ and $ q $ and determines the truth values of the compound statements.

| $ p $ | $ q $ | $ p \rightarrow q $ | $ q \rightarrow p $ | $ p \vee q $ | Both premises true? | Conclusion true if premises true? |
|:------:|:------:|:--------------------:|:--------------------:|:-------------:|:------------------:|:------------------------------:|
| T | T | T | T | T | True | True |
| T | F | F | T | T | False | True |
| F | T | T | F | T | False | True |
| F | F | T | T | F | True | False |

3. **Analysis**:
- When both $ p $ and $ q $ are true (first row), $ p \rightarrow q $ is true, $ q \rightarrow p $ is true, and $ p \vee q $ is true. Both the

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