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Decide the truth values of the following statement: (~r^s) ⇒ (~s ⇒ (q V r)) by filling in the truth table below. qrs Is the statement in (a) a tautology? Justify your answer. Is the statement in (a) logically equivalent to (q V r)? Justify your answer.

Decide the truth values of the following statement: (~r^s) ⇒ (~s ⇒ (q V r)) by filling in the truth table below. qrs Is the statement in (a) a tautology? Justify your answer. Is the statement in (a) logically equivalent to (q V r)? Justify your answer.

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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### Exercise on Logical Equivalence and Tautologies (a) **Decide the truth values of the following statement: \((\neg r \land s) \implies (\neg s \iff (q \lor r))\) by filling in the truth table below.** | q | r | s | ¬r | ¬s | ¬r ∧ s | q ∨ r | ¬s ⇔ (q ∨ r) | (¬r ∧ s) ⇒ (¬s ⇔ (q ∨ r)) | |:-:|:-:|:-:|:----:|:----:|:--------:|:--------:|:------------:|:--------------------------:| | T | T | T | F | F | F | T | F | T | | T | T | F | F | T | F | T | T | T | | T | F | T | T | F | T | T | F | F | | T | F | F | T | T | F | T | T | T | | F | T | T | F | F | F | T | F | T | | F | T | F | F | T | F | T | T | T | | F | F | T | T | F | T | F | T | T | | F | F | F | T | T | F | F | F | T | (b) **Is the statement in (a) a tautology? Justify your answer.** **Answer:** A statement is a tautology if it is always true regardless of the truth values of its components. From the completed truth table, we see that the statement \((\neg r \land s) \implies (\neg s \iff (q \lor r))\) is true in all possible cases except one (
Transcribed Image Text:### Exercise on Logical Equivalence and Tautologies (a) **Decide the truth values of the following statement: \((\neg r \land s) \implies (\neg s \iff (q \lor r))\) by filling in the truth table below.** | q | r | s | ¬r | ¬s | ¬r ∧ s | q ∨ r | ¬s ⇔ (q ∨ r) | (¬r ∧ s) ⇒ (¬s ⇔ (q ∨ r)) | |:-:|:-:|:-:|:----:|:----:|:--------:|:--------:|:------------:|:--------------------------:| | T | T | T | F | F | F | T | F | T | | T | T | F | F | T | F | T | T | T | | T | F | T | T | F | T | T | F | F | | T | F | F | T | T | F | T | T | T | | F | T | T | F | F | F | T | F | T | | F | T | F | F | T | F | T | T | T | | F | F | T | T | F | T | F | T | T | | F | F | F | T | T | F | F | F | T | (b) **Is the statement in (a) a tautology? Justify your answer.** **Answer:** A statement is a tautology if it is always true regardless of the truth values of its components. From the completed truth table, we see that the statement \((\neg r \land s) \implies (\neg s \iff (q \lor r))\) is true in all possible cases except one (
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