(a) Complete the following truth table. Use T for true and F for false. You may add more columns, but those added columns will not be graded. P 9 TT T F FT LL F F ~(pvq)^(p^~qg) 10 0 0 10 Р ~O ローロ X 9 A ローロ OVO S hs
(a) Complete the following truth table. Use T for true and F for false. You may add more columns, but those added columns will not be graded. P 9 TT T F FT LL F F ~(pvq)^(p^~qg) 10 0 0 10 Р ~O ローロ X 9 A ローロ OVO S hs
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 if a Statement is a Tautology, Contradiction, or Neither
**Instructions:**
1. Complete the following truth table. Use T for true and F for false. You may add more columns, but those added columns will not be graded.
**Given Statement:**
~(p ∨ q) ∧ (p ∧ ~q)
#### Truth Table:
| p | q | ~(p ∨ q) | p ∧ ~q | ~(p ∨ q) ∧ (p ∧ ~q) |
|---|---|---------|-----|------------------------|
| T | T | F | F | F |
| T | F | F | T | F |
| F | T | F | F | F |
| F | F | T | F | F |
(b) Is the statement ~(p ∨ q) ∧ (p ∧ ~q) a tautology, a contradiction, or neither? Why? Choose the best answer.
- The statement is a tautology. This is because it is true for all possible true-false combinations of p and q.
- The statement is a contradiction. This is because it is false for every possible combination of p and q.
- The statement is neither a tautology nor a contradiction. This is since it is true for some combinations of p and q and false for others.
**Answer:**
- The statement is a contradiction. This is because it is false for every possible combination of p and q.
**Explanation:**
The truth table shows that no matter what values of p and q are chosen (both true, both false, one true and one false), the resulting expression ~(p ∨ q) ∧ (p ∧ ~q) always evaluates to false, hence it is a contradiction.
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---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff452152e-984b-4f04-b158-1202c867fcfb%2Fffa4ace7-0c80-4b35-ba6b-a56a3cb8aeb5%2Fa7s2cyf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Determining if a Statement is a Tautology, Contradiction, or Neither
**Instructions:**
1. Complete the following truth table. Use T for true and F for false. You may add more columns, but those added columns will not be graded.
**Given Statement:**
~(p ∨ q) ∧ (p ∧ ~q)
#### Truth Table:
| p | q | ~(p ∨ q) | p ∧ ~q | ~(p ∨ q) ∧ (p ∧ ~q) |
|---|---|---------|-----|------------------------|
| T | T | F | F | F |
| T | F | F | T | F |
| F | T | F | F | F |
| F | F | T | F | F |
(b) Is the statement ~(p ∨ q) ∧ (p ∧ ~q) a tautology, a contradiction, or neither? Why? Choose the best answer.
- The statement is a tautology. This is because it is true for all possible true-false combinations of p and q.
- The statement is a contradiction. This is because it is false for every possible combination of p and q.
- The statement is neither a tautology nor a contradiction. This is since it is true for some combinations of p and q and false for others.
**Answer:**
- The statement is a contradiction. This is because it is false for every possible combination of p and q.
**Explanation:**
The truth table shows that no matter what values of p and q are chosen (both true, both false, one true and one false), the resulting expression ~(p ∨ q) ∧ (p ∧ ~q) always evaluates to false, hence it is a contradiction.
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© 2023 McGraw Hill LLC. All Rights Reserved.
---
![### Determining if a Statement is a Tautology, Contradiction, or Neither
In logic, it is important to determine whether a given statement is a tautology, a contradiction, or neither. This understanding helps in evaluating logical arguments and constructing proofs.
#### Example Problem:
**Given Statement:**
Determine if the following statement is a tautology, a contradiction, or neither, and explain why:
\[(p \vee q) \land (p \land \neg q)\]
#### Options:
1. **The statement is a tautology. This is because it is true for all possible true-false combinations of \(p\) and \(q\).**
2. **The statement is a tautology. This is because it is true for some true-false combinations of \(p\) and \(q\) and false for others.**
3. **The statement is a contradiction. This is because it is false for all possible true-false combinations of \(p\) and \(q\).**
4. **The statement is a contradiction. This is because it is true for some true-false combinations of \(p\) and \(q\) and false for others.**
5. **The statement is neither a tautology nor a contradiction.**
#### Explanation:
- **Tautology:** A statement that is always true regardless of the truth values of the propositions involved.
- **Contradiction:** A statement that is always false regardless of the truth values of the propositions involved.
- **Neither:** A statement that is true for some truth values and false for others.
#### Graphs and Diagrams:
1. **Truth Table Construction:**
- **Step 1:** Evaluate \(p \vee q\)
- **Step 2:** Evaluate \(p \land \neg q\)
- **Step 3:** Combine the results using \((p \vee q) \land (p \land \neg q)\)
| \(p\) | \(q\) | \(\neg q\) | \(p \vee q\) | \(p \land \neg q\) | \((p \vee q) \land (p \land \neg q)\) |
|------|------|-------------|-------------|-------------------|------------------------------------|
| T | T | F | T | F | F |
| T | F | T](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff452152e-984b-4f04-b158-1202c867fcfb%2Fffa4ace7-0c80-4b35-ba6b-a56a3cb8aeb5%2Fevk632c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Determining if a Statement is a Tautology, Contradiction, or Neither
In logic, it is important to determine whether a given statement is a tautology, a contradiction, or neither. This understanding helps in evaluating logical arguments and constructing proofs.
#### Example Problem:
**Given Statement:**
Determine if the following statement is a tautology, a contradiction, or neither, and explain why:
\[(p \vee q) \land (p \land \neg q)\]
#### Options:
1. **The statement is a tautology. This is because it is true for all possible true-false combinations of \(p\) and \(q\).**
2. **The statement is a tautology. This is because it is true for some true-false combinations of \(p\) and \(q\) and false for others.**
3. **The statement is a contradiction. This is because it is false for all possible true-false combinations of \(p\) and \(q\).**
4. **The statement is a contradiction. This is because it is true for some true-false combinations of \(p\) and \(q\) and false for others.**
5. **The statement is neither a tautology nor a contradiction.**
#### Explanation:
- **Tautology:** A statement that is always true regardless of the truth values of the propositions involved.
- **Contradiction:** A statement that is always false regardless of the truth values of the propositions involved.
- **Neither:** A statement that is true for some truth values and false for others.
#### Graphs and Diagrams:
1. **Truth Table Construction:**
- **Step 1:** Evaluate \(p \vee q\)
- **Step 2:** Evaluate \(p \land \neg q\)
- **Step 3:** Combine the results using \((p \vee q) \land (p \land \neg q)\)
| \(p\) | \(q\) | \(\neg q\) | \(p \vee q\) | \(p \land \neg q\) | \((p \vee q) \land (p \land \neg q)\) |
|------|------|-------------|-------------|-------------------|------------------------------------|
| T | T | F | T | F | F |
| T | F | T
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