Use a truth table to determine whether the two statements are equivalent. (~p→q)^(q→~p) and q-p q ~p→q)^(q→~p) q-p 306562 SOSA RUHAZ verdthalie H mesta, da tabe mes INDIG ZDA! F POMINA RESIDE T F F Choose the correct answer below. O The statements are equivalent. O The statements are not equivalent. Complete the truth table. PT p T T T LL FL
Use a truth table to determine whether the two statements are equivalent. (~p→q)^(q→~p) and q-p q ~p→q)^(q→~p) q-p 306562 SOSA RUHAZ verdthalie H mesta, da tabe mes INDIG ZDA! F POMINA RESIDE T F F Choose the correct answer below. O The statements are equivalent. O The statements are not equivalent. Complete the truth table. PT p T T T LL FL
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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Transcribed Image Text:### Logical Equivalence and Truth Tables
**Instruction:**
Use a truth table to determine whether the two statements are equivalent: \((\neg p \rightarrow q) \land (q \rightarrow \neg p)\) and \(q \leftrightarrow \neg p\).
**Truth Table:**
| \(p\) | \(q\) | \(\neg p\) | \(\neg p \rightarrow q\) | \(q \rightarrow \neg p\) | \((\neg p \rightarrow q) \land (q \rightarrow \neg p)\) | \(q \leftrightarrow \neg p\) |
|:----:|:----:|:---------:|:-----------------------:|:-----------------------:|:-----------------------------------------------------:|:-------------------------:|
| T | T | F | T | F | F | F |
| T | F | F | T | T | T | T |
| F | T | T | T | T | T | T |
| F | F | T | F | T | F | F |
### Analysis:
- **Column \( \neg p \)**: Represents the negation of \( p \).
- **Columns \( \neg p \rightarrow q \) and \( q \rightarrow \neg p \)**: Represents conditional statements.
- **Column \((\neg p \rightarrow q) \land (q \rightarrow \neg p)\)**: Represents the conjunction of the two conditional statements.
- **Column \( q \leftrightarrow \neg p \)**: Represents the biconditional statement.
### Conclusion:
The truth values in columns \((\neg p \rightarrow q) \land (q \rightarrow \neg p)\) and \( q \leftrightarrow \neg p \) are identical for all possible truth values of \( p \) and \( q \). Hence, the statements are equivalent.
**Choose the correct answer below:**
- \( \textcircled{O} \) The statements are equivalent.
- \( \) The statements are not equivalent.
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