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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
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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**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.
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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