TF r F F T If P, then Q is logically equivalent to If ~P, then ~Q. The statement ~P --> ~Q is the converse of P --> Q. The negation of ~P --> Q is ~PA~Q. The statement D is ogu lont to RV
TF r F F T If P, then Q is logically equivalent to If ~P, then ~Q. The statement ~P --> ~Q is the converse of P --> Q. The negation of ~P --> Q is ~PA~Q. The statement D is ogu lont to RV
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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Question
Help with true or false for following
Transcribed Image Text:**Logical Equivalences and Mathematical Statements**
This document provides a series of statements evaluated for their truthfulness. Each entry features a truth value (T/F) followed by a logical or mathematical assertion.
1. **Logical Equivalence:**
- **Statement:** If P, then Q is logically equivalent to If ~P, then ~Q.
- **Evaluation:** False
2. **Converse Statement:**
- **Statement:** The statement ~P --> ~Q is the converse of P --> Q.
- **Evaluation:** False
3. **Negation:**
- **Statement:** The negation of ~P --> Q is ~P ∧ ~Q.
- **Evaluation:** False
4. **Logical Statement:**
- **Statement:** The statement P --> Q is equivalent to ~P ∨ Q.
- **Evaluation:** True
5. **Proof by Contradiction:**
- **Statement:** To prove the statement: If ab = 0 then a = 0 or b = 0, you may assume ab = 0 and a ≠ 0 and then deduce that b = 0.
- **Evaluation:** True
6. **Modulo Operation:**
- **Statement:** If a ≡ 4 (mod 8), then a² ≡ 0 (mod 8).
- **Evaluation:** False
7. **Modulo Operation:**
- **Statement:** If a² ≡ 1 (mod 8), then a ≡ 1 (mod 8).
- **Evaluation:** False
8. **Integer Property:**
- **Statement:** For all integers n, 2 | (n⁴ + n).
- **Evaluation:** True
9. **Last Digit of a Power:**
- **Statement:** The last digit of 6⁴⁰⁰ is a six.
- **Evaluation:** True
10. **Last Two Digits of a Power:**
- **Statement:** The last two digits of 5¹³³ are 33.
- **Evaluation:** False
This collection serves as an educational resource for understanding logical statements and conducting mathematical proofs, including equivalences, negations, and properties of integers under modular arithmetic.
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