course Mth 151
7/12 12
Question: `q001. There are 9 questions in this set.
Explain why [ (p -> q) ^ (q -> r) ^ p] -> r must be true for every set of truth values for which r is true.
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Your solution:
This is true because of what r represents which is “will do” for instance “Rita will cook” which will have to make p to q be true and q to r be true because r being T means everything with R must also be true. R makes it true of false.
2
Should have drawn a table.
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Question: `q002. At this point we know that the truth values TTT, TFT, FTT, FFT all make the argument [ (p -> q) ^ (q -> r) ^ p] -> r true. What about the truth values TTF?
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Your solution:
It would make the assessment false because r is false
2
Ok
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Question: `q003. The preceding statement said that for the TTF case [ (p -> q) ^ (q -> r) ^ p] was false but did not provide an explanation of this statement. Which of the statements is false for the truth values TTF, and what does this tell us about the truth of the statement [ (p -> q) ^ (q -> r) ^ p] -> r?
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Your solution:
The TTF makes the last statement False so that would be q-> r which makes the entire statement must be false because r is false
2
Ok
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Question: `q004. Examine the truth of the statement [ (p -> q) ^ (q -> r) ^ p] for each of the truth sets TFF, FTF and FFF.
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Your solution:
In the first one p is true and q is false making the statement false
Second one p is false and q is true so the statement would be false as well
Third p is false making the statement false again
2
Ok
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Question: `q005. We have seen that for TFF, FTF and FFF the statement [ (p -> q) ^ (q -> r) ^ p] is false. How does this help us establish that [ (p -> q) ^ (q -> r) ^ p] -> r is always true?
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Your solution:
The ones we looked at r was false therefore in the case true anytime r is false
2
Ok
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Question: `q006. Explain how we have shown in the past few exercises that [ (p -> q) ^ (q -> r) ^ p] -> r must always be true.
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Your solution:
We showed that when r is true anything with r in it must be true and when r is false the statement must be true so r is always true
3
Ok
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Question: `q007. Explain how this shows that the original argument about rain, wet grass and smelling wet grass, must be valid.
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Your solution:
The statement is always true so it must be valid
3
Ok
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Question: `q008. Explain how the conclusion of the last example, that [ (p -> q) ^ (q -> r) ^ p] -> r is always a true statement, shows that the following argument is valid: 'If it snows, the roads are slippery. If the roads are slippery they'll be safer to drive on. It just snowed. Therefore the roads are safer to drive on.' Hint: First symbolize the present argument.
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Your solution:
Let P stand for snow and Q stand for when it snows and R for safer to drive on
P-> Q stands for if it snows the roads are slippery
Q-> R is if the roads are slippery they will be easier to drive on
The argument is if [p -> q, and q -> r, and p] are all true then r is true
This is [ (p -> q) ^ (q -> r) ^ p] -> r. the argument is valid because this is the same as before and we have shown it is always true
3
Ok
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Question: `q009. Symbolize the following argument and show that it is valid: 'If it doesn't rain there is a picnic. There is no picnic. Therefore it rained.'
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Your solution:
P stands for it rained q there is a picnic. First statement If it doesn’t rain there is a picnic would be ~p -> q. The second statement There is no picnic stands for ~q. The conclusion it rained is symbolized by p. The argument says if [ (~p -> q) and ~q] then p the truth table will show
p q ~p ~q ~p -> q (~p -> q) ^ ~q [ (~p -> q) ^ ~q ] -> p
T T F F T F T
T F F T T T T
F T T F T F T
F F T T F F T
2
ok
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