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A type of proof we have not used yet is the proof by cases. If we want to prove that p → q, it is at times convenient to rewrite p as a disjunction of simpler propositions, i.e., p = p1 V P2 V ... V Pn and then show that (p1 V p2 V ... V Pn) → q. This last implication, although apparently more complex, is often simpler because of the following tautology: [(P1 V p2 V... V Pn) → q] → [(p1 → q) ^ (p2 → q) ^...A (Pn → q)] Here, each of the pi → q is typically simpler to prove than p → q. Prove the relationship above is a tautology (to keep things simple, let n = 3, although the result holds for an arbitrary n.)

A type of proof we have not used yet is the proof by cases. If we want to prove that p → q, it is at times convenient to rewrite p as a disjunction of simpler propositions, i.e., p = p1 V P2 V ... V Pn and then show that (p1 V p2 V ... V Pn) → q. This last implication, although apparently more complex, is often simpler because of the following tautology: [(P1 V p2 V... V Pn) → q] → [(p1 → q) ^ (p2 → q) ^...A (Pn → q)] Here, each of the pi → q is typically simpler to prove than p → q. Prove the relationship above is a tautology (to keep things simple, let n = 3, although the result holds for an arbitrary n.)

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 type of proof we have not used yet is the proof by cases. If we want to prove that p → q, it is at times convenient to rewrite p as a disjunction of simpler propositions, i.e., p = p1 V p2 V ... V Pn and then show that (p1 V p2 V ... V pn) → q. This last implication, although apparently more complex, is often simpler because of the following tautology: [(P1 V p2 V ... V Pn) → q] + [(P1 → q) ^ (p2 → q) ^...A (Pn → q)] Here, each of the pi → q is typically simpler to prove than p→ q. Prove the relationship above is a tautology (to keep things simple, let n = 3, although the result holds for an arbitrary n.)
Transcribed Image Text:A type of proof we have not used yet is the proof by cases. If we want to prove that p → q, it is at times convenient to rewrite p as a disjunction of simpler propositions, i.e., p = p1 V p2 V ... V Pn and then show that (p1 V p2 V ... V pn) → q. This last implication, although apparently more complex, is often simpler because of the following tautology: [(P1 V p2 V ... V Pn) → q] + [(P1 → q) ^ (p2 → q) ^...A (Pn → q)] Here, each of the pi → q is typically simpler to prove than p→ q. Prove the relationship above is a tautology (to keep things simple, let n = 3, although the result holds for an arbitrary n.)
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