Use a truth table to determine whether the following is a tautology, a contradic- tion, or neither. (a) (PVQ) ^ (~ P^~Q) (b) P⇒ [(~ P) ⇒ (Q^~Q)]
Use a truth table to determine whether the following is a tautology, a contradic- tion, or neither. (a) (PVQ) ^ (~ P^~Q) (b) P⇒ [(~ P) ⇒ (Q^~Q)]
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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Show your work and all steps please.
![**Determine Tautology, Contradiction, or Neither Using a Truth Table**
To determine whether the following propositions are tautologies, contradictions, or neither, we will use truth tables.
**(a) \( (P \lor Q) \land (\neg P \land \neg Q) \)**
We need to construct a truth table to evaluate this compound proposition. Here is the breakdown of the steps required:
1. List all possible truth values combinations for the basic propositions \( P \) and \( Q \).
2. Compute \( P \lor Q \).
3. Compute \( \neg P \) and \( \neg Q \).
4. Compute \( \neg P \land \neg Q \).
5. Finally, compute the resultant \( (P \lor Q) \land (\neg P \land \neg Q) \).
Format for truth table:
| \( P \) | \( Q \) | \( P \lor Q \) | \( \neg P \) | \( \neg Q \) | \( \neg P \land \neg Q \) | \( (P \lor Q) \land (\neg P \land \neg Q) \) |
|:------:|:------:|:-------------:|:----------:|:----------:|:----------------------:|:-----------------------------:|
| T | T | T | F | F | F | F |
| T | F | T | F | T | F | F |
| F | T | T | T | F | F | F |
| F | F | F | T | T | T | F |
In each row, the final column \( (P \lor Q) \land (\neg P \land \neg Q) \) is false (F), indicating that this proposition is a contradiction.
**(b) \( P \leftrightarrow [(\neg P) \Rightarrow (Q \land \neg Q)] \)**
We need to construct a truth table to evaluate this compound proposition. Here is the breakdown of the steps required:
1. List all possible truth values combinations for the basic propositions \( P \) and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F47a2ecfa-9432-49fb-bf43-e148a3a9aff5%2Fab8f35a1-ff98-45a3-8ff3-c9167089f6bb%2Ft1niymn_processed.png&w=3840&q=75)
Transcribed Image Text:**Determine Tautology, Contradiction, or Neither Using a Truth Table**
To determine whether the following propositions are tautologies, contradictions, or neither, we will use truth tables.
**(a) \( (P \lor Q) \land (\neg P \land \neg Q) \)**
We need to construct a truth table to evaluate this compound proposition. Here is the breakdown of the steps required:
1. List all possible truth values combinations for the basic propositions \( P \) and \( Q \).
2. Compute \( P \lor Q \).
3. Compute \( \neg P \) and \( \neg Q \).
4. Compute \( \neg P \land \neg Q \).
5. Finally, compute the resultant \( (P \lor Q) \land (\neg P \land \neg Q) \).
Format for truth table:
| \( P \) | \( Q \) | \( P \lor Q \) | \( \neg P \) | \( \neg Q \) | \( \neg P \land \neg Q \) | \( (P \lor Q) \land (\neg P \land \neg Q) \) |
|:------:|:------:|:-------------:|:----------:|:----------:|:----------------------:|:-----------------------------:|
| T | T | T | F | F | F | F |
| T | F | T | F | T | F | F |
| F | T | T | T | F | F | F |
| F | F | F | T | T | T | F |
In each row, the final column \( (P \lor Q) \land (\neg P \land \neg Q) \) is false (F), indicating that this proposition is a contradiction.
**(b) \( P \leftrightarrow [(\neg P) \Rightarrow (Q \land \neg Q)] \)**
We need to construct a truth table to evaluate this compound proposition. Here is the breakdown of the steps required:
1. List all possible truth values combinations for the basic propositions \( P \) and
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