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(a) If we proved P⇒ Q and we know that is true, then we can conclude that P is true. ((P⇒Q)^Q) ⇒ P. (b) If we proved P⇒ Q and we know that P is false, then we can conclude that is false. ((PQ) ^¬P) ⇒ ¬Q. (c) If we proved P⇒ Q and we know that Q is false, then we can conclude that P is false. ((PQ) ^¬Q) ⇒ ¬P.

(a) If we proved P⇒ Q and we know that is true, then we can conclude that P is true. ((P⇒Q)^Q) ⇒ P. (b) If we proved P⇒ Q and we know that P is false, then we can conclude that is false. ((PQ) ^¬P) ⇒ ¬Q. (c) If we proved P⇒ Q and we know that Q is false, then we can conclude that P is false. ((PQ) ^¬Q) ⇒ ¬P.

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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(a) If we proved P⇒ Q and we know that is true, then we can conclude that P is true. ((P = ⇒>> Q) ^Q) ⇒ P. (b) If we proved P⇒ Q and we know that P is false, then we can conclude that is false. ((P⇒ Q) ^¬P) ⇒ ¬Q. (c) If we proved P⇒ Q and we know that Q is false, then we can conclude that P is false. ((PQ)^¬Q) ⇒ ¬P.
Transcribed Image Text:(a) If we proved P⇒ Q and we know that is true, then we can conclude that P is true. ((P = ⇒>> Q) ^Q) ⇒ P. (b) If we proved P⇒ Q and we know that P is false, then we can conclude that is false. ((P⇒ Q) ^¬P) ⇒ ¬Q. (c) If we proved P⇒ Q and we know that Q is false, then we can conclude that P is false. ((PQ)^¬Q) ⇒ ¬P.
en argument is valid if the corresponding logical formula is a tautology (that is, its truth table has a T in every row). For example, suppose you have specific P, Q for which you know that P⇒ Q is true, and you also know that P is true. Then you can conclude that Q must be true¹, because the logical formula ((P⇒Q) ^ P) ⇒ Q is valid (a tautology). Determine which of the following arguments are valid by checking if the given logical formula is a tautology. You can either use truth tables, or you can use the fact that an implication A B is false only if A is true and B is false.
Transcribed Image Text:en argument is valid if the corresponding logical formula is a tautology (that is, its truth table has a T in every row). For example, suppose you have specific P, Q for which you know that P⇒ Q is true, and you also know that P is true. Then you can conclude that Q must be true¹, because the logical formula ((P⇒Q) ^ P) ⇒ Q is valid (a tautology). Determine which of the following arguments are valid by checking if the given logical formula is a tautology. You can either use truth tables, or you can use the fact that an implication A B is false only if A is true and B is false.
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