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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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

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

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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Question

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 4.4
Determine 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 invalid
argument. Having a cold is not the only cause of sleeping difficulty. It may be caused by some
other factors.
Further, we may consider an argument with the following symbolic form.
q-r
Premise 1
r-s
Premise 2
-t-s
Premise 3
Premise 4
9
To determine whether the argument is valid or invalid using truth table would require a
table with 2* = 16 rows. It would be time consuming to construct such table and with the large
number of truth values to be determined, we might make an error. Thus we consider a different
approach that makes use of a sequence of valid arguments to arrive at a conclusion.
Consider the above example.
Premise 1
Premise 2
Transitive Reasoning
q→r
ris
:.q-s
q-s
s t
9-t
q-t
9
at
The sequence of a valid arguments shows that is a valid conclusion for the original
argument. Thus the original argument is a valid argument.
Example 4.4
Determine 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's
production costs must exceed 50 million dollars (c) or I do not want to see it. The movie's
production costs were less than 50 million dollars. Therefore, the movie was not directed by
Steven Spielrberg.
Solution:
S-W
CV-W
Previous Conclusion
Premise 3
Transitive Reasoning
Transforming the argument to its symbolic form yields
Premise 1
Premise 2
Premise 3
Conclusion
. ~S
Previous Conclusion
Premise 4
Direct Reasoning
S W
WIC
Premise 2 can be written as w→ cas previously discussed. Applying transitive reasoning and
this equivalent form of Premise 2 produces
SIC
Premise 1
Premise 2
Transitive Reasoning

Combining the conclusion s→c with Premise 3 gives us
Previous Conclusion
Premise 3
Contrapositive Reasoning
SIC
~C
:~S
The 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 fall
asleep. If I drink soda then I must eat candy bar (c). Therefore, I eat candy bar whenever I read
a Math book.
Solution:
Transforming the argument in its symbolic form yields
m-a
a s
SIC
:m-c
Combining Premise 1 and Premise 2 results to
Premise 1
Premise 2
Transitive Reasoning
Next we combine the previous conclusion with Premise 3
m-a
a s
m-s
Premise 1
Premise 2
Premise 3
Conclusion
m-s
SIC
:m-c
Previous Conclusion
Premise 3
Transitive Reasoning
We have shown that the process resulted to desired conclusion, mc, hence, the argument is
valid.

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