In exercises 11-12, use all the premises to determine the valid conclusion for the given argument.1. If it is a theropod, then it is not herbivorous. If it is not herbivorous, then it is not sauropod. It is a sauropod. Therefore,2. If you buy the car, you will need a loan. You do not need a loan or you will make a monthly payments. You buy the car. Therefore,Important note: please make solutions TYPEWRITTEN and NOT HANDWRITTEN Use the given lesson attached as 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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