In exercises 7-10, use a sequence of valid arguments to show that each argument is valid.7. If we sell the boat (b), then we will not go to the river (~r). If we don't go to the river, then we will go to camping (c). If we do not buy a tent (~t), then we will not go to camping. Therefore, if we sell the boat then we will buy a tent.8. If it is an ammonite (a), then it is from cretaceous period (c). If it is not from Mesozoic era (~m), then it is not from the cretaceous period. It is from Mesozoic era, then it is at least 65 million years old (s). Therefore, if it is an ammonite, then it is at least 65 million years old.9. If the computer is not operating (~0), then I will not be able to finish my report (~f). If the office is closed (c), then the computer is not operating. Therefore, if I am able to finish my report, then the office is open.10. If he reads the manuscript (r), he will like it. (1) If he likes it, he will publish it (p). If he publishes it, then you will get royalties (m). You did not get royalties. Therefore, he did not read the manuscript.NOTE: please refer to the given lesson for reference Example 4.4Determine whether the argument is valid or invalid.1. If I had a cold, then I find it difficult to sleep.I find it difficult to sleep.Therefore, I have a cold.Solution: This matches the form known as fallacy of the converse. Thus, this is an invalidargument. Having a cold is not the only cause of sleeping difficulty. It may be caused by someother factors.Further, we may consider an argument with the following symbolic form.q-rPremise 1r-sPremise 2-t-sPremise 3Premise 49To determine whether the argument is valid or invalid using truth table would require atable with 2* = 16 rows. It would be time consuming to construct such table and with the largenumber of truth values to be determined, we might make an error. Thus we consider a differentapproach that makes use of a sequence of valid arguments to arrive at a conclusion.Consider the above example.Premise 1Premise 2Transitive Reasoningq→rris:.q-sq-ss t9-tq-t9atThe sequence of a valid arguments shows that is a valid conclusion for the originalargument. Thus the original argument is a valid argument.Example 4.4Determine whether the following arguments are valid.1. If the movie was directed by Steven Spielberg (s), then I want to see it (w). The movie'sproduction costs must exceed 50 million dollars (c) or I do not want to see it. The movie'sproduction costs were less than 50 million dollars. Therefore, the movie was not directed bySteven Spielrberg.Solution:S-WCV-WPrevious ConclusionPremise 3Transitive ReasoningTransforming the argument to its symbolic form yieldsPremise 1Premise 2Premise 3Conclusion. ~SPrevious ConclusionPremise 4Direct ReasoningS WWICPremise 2 can be written as w→ cas previously discussed. Applying transitive reasoning andthis equivalent form of Premise 2 producesSICPremise 1Premise 2Transitive Reasoning Combining the conclusion s→c with Premise 3 gives usPrevious ConclusionPremise 3Contrapositive ReasoningSIC~C:~SThe sequence has produced the desired conclusion. Thus, the original argument is valid.2. I start to fall asleep (a) if I read a Math book (m). I drink soda (s) whenever I start to fallasleep. If I drink soda then I must eat candy bar (c). Therefore, I eat candy bar whenever I reada Math book.Solution:Transforming the argument in its symbolic form yieldsm-aa sSIC:m-cCombining Premise 1 and Premise 2 results toPremise 1Premise 2Transitive ReasoningNext we combine the previous conclusion with Premise 3m-aa sm-sPremise 1Premise 2Premise 3Conclusionm-sSIC:m-cPrevious ConclusionPremise 3Transitive ReasoningWe have shown that the process resulted to desired conclusion, mc, hence, the argument isvalid.

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10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ