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
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.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F406074af-a425-40d0-923c-17adedf14fc2%2F63435fac-bd8e-4b1c-8bad-1a90084577d6%2Fs3gy5bv_processed.png&w=3840&q=75)
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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