#$&* course MTH 151 10:23pm, 3/25/14 If your solution to stated problem does not match the given solution, you should self-critique per instructions at
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Given Solution: [ (p -> q) ^ (q -> r) ^ p] -> r must be true if r is true, since the statement can only be false if [ (p -> q) ^ (q -> r) ^ p] is true while r is false. Therefore the truth values TTT, TFT, FTT, FFT (i.e., all the truth values that have r true) all make the statement true. You could make a table, which would be useful in understanding the above explanation. STUDENT COMMENT: I still don't quite grasp this. Is this the same thing as the table? INSTRUCTOR RESPONSE: On any question where you don't understand the given solution, you should break the given explanation up into phrases and tell me what you do and do not understand about each. For example, on this problem you might break the explanation up as follows: [ (p -> q) ^ (q -> r) ^ p] -> r must be true if r is true (Do you understand what this is saying? Do you understand why it must be so? Exactly what do you understand and what do you not understand about this statement?) the statement can only be false if [ (p -> q) ^ (q -> r) ^ p] is true while r is false. (Do you understand what this is saying? Do you understand why it must be so? Exactly what do you understand and what do you not understand about this statement?) TTT, TFT, FTT, FFT are all the truth values that have r true (Do you understand what this is saying? Do you understand why it must be so? Exactly what do you understand and what do you not understand about this statement?) the truth values TTT, TFT, FTT, FFT (i.e., all the truth values that have r true) all make the statement true. (Do you understand what this is saying? Do you understand why it must be so? Exactly what do you understand and what do you not understand about this statement?) [ (p -> q) ^ (q -> r) ^ p] -> r must be true if r is true, since the statement can only be false if [ (p -> q) ^ (q -> r) ^ p] is true while r is false. (Do you understand what this is saying? Do you understand why it must be so? Exactly what do you understand and what do you not understand about this statement?) Now putting it all together: [ (p -> q) ^ (q -> r) ^ p] -> r must be true if r is true, since the statement can only be false if [ (p -> q) ^ (q -> r) ^ p] is true while r is false. Therefore the truth values TTT, TFT, FTT, FFT (i.e., all the truth values that have r true) all make the statement true. (Do you understand what this is saying? Do you understand why it must be so? Exactly what do you understand and what do you not understand about this statement?) STUDENT COMMENT: so r is the term that makes it true or false ------------------------------------------------ Self-critique Rating:ent: 3 INSTRUCTOR RESPONSE: The consequent r does by itself does not necessarily determine the truth of the statement. If r is true, then the statement is true. However if r is false then the statement might be true or false. If the conclusion r is false, then if the antecedent (in this case [ (p -> q) ^ (q -> r) ^ p]) is true the statement is false. However if the antecedent is false, then the statement is true, despite the fact that r is false. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I’m still not understanding this, even after reading your solution. I’ve never been great at math, but usually I can understand most basic concepts after a little work and studying, and reviewing the problems. I have no idea what the problem is with this, I am simply not understanding it in any way. ------------------------------------------------ Self-critique Rating: 3 ********************************************* 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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Although I feel unsure about my answer, if I had to guess, I would say that TTF would make that argument false, because TTF would be the opposite of FFT, and FFT was true, so it would seem like TTF would be false. confidence rating #$&*: 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: It would be possible to evaluate every one of the statements p -> q, q -> r, etc. for their truth values, given truth values TTF. However we can shortcut the process. We see that [ (p -> q) ^ (q -> r) ^ p] is a compound statement with conjunction ^. This means that [ (p -> q) ^ (q -> r) ^ p] will be false if any one of the three compound statements p -> q, q -> r, p is false. For TTF we see that one of these statements is false, so that [ (p -> q) ^ (q -> r) ^ p] is false. This therefore makes the statement [ (p -> q) ^ (q -> r) ^ p] -> r true. STUDENT COMMENT I’m still confused as to why one statement being false makes the entire situation true, but I think I will figure it out. INSTRUCTOR RESPONSE If the statement is A -> B, then if A is false, the statement is true. This is because the only way for A -> B to be false is for A to be true and B false. Since A isn't false, A -> B isn't false, and if a statement isn't false, then (since it must be true or false) it must be true. Now A would be the compound statement (p -> q) ^ (q -> r) ^ p. Since this is a string of ^ statements, it can only be true of all the statments p -> q, q -> r and p are true. Since p isn't true, the string of ^ statements isn't true, so it must be false. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating: 3 ********************************************* 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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I’m just not understanding the question, and am not sure of how to make a guess towards the answer. confidence rating #$&*: 0 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: p and q are both true, so p -> q and p are true. The only candidate for a false statement among the three statements is q -> r. So we evaluate q -> r for truth values TTF. Since q is T and r is F, we see that q -> r must be F. This makes [ (p -> q) ^ (q -> r) ^ p] false. Therefore [ (p -> q) ^ (q -> r) ^ p] -> r must be true, since it can only be false and if [ (p -> q) ^ (q -> r) ^ p] is true. STUDENT QUESTION Explain to me about finding truth in these sets such as TTF. I can't find it in the book nor did the lady on the video say anything about them. INSTRUCTOR RESPONSE TFF stands for the truth values of p, q and r. TFF means the p is true, while q and r are both false. In your truth table this corresponds to the fourth line, which should read: p q r p->q q->r [(p->q)^(q->r)^p] [(p->q)^(q->r)^p] [(p->q)^(q->r)^p] [(p->q)^(q->r)^p]->r T T F T F F T Note that [(p->q)^(q->r)^p] is false for this line, because this expression is a conjunction and at least one of the statement s in the conjunction is false. This makes [(p->q)^(q->r)^p] - r true, since a false antecedent makes the conditional true. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q004. Examine the truth of the statement [ (p -> q) ^ (q -> r) ^ p] for each of the truth sets TFF, FTF and FFF. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I have tried to understand this, and just can’t come up with any answer that I feel like would even be close to being right. I just don’t even know what to try and guess. confidence rating #$&*: 0 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: In the case TFF, p is true and q is false so p -> q is false, which makes [ (p -> q) ^ (q -> r) ^ p] false. In the case FTF, p is false, making [ (p -> q) ^ (q -> r) ^ p] false. In the case FFF, p is again false, making [ (p -> q) ^ (q -> r) ^ p] false. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating: ********************************************* 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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The only reason I could see how you could see how they both would always be true is because they both contain the same argument, [ (p -> q) ^ (q -> r) ^ p]. confidence rating #$&*: 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: The three given truth values, plus the TTF we examined earlier, are all the possibilities where r is false. We see that in the cases where r is false, [ (p -> q) ^ (q -> r) ^ p] is always false. This makes [ (p -> q) ^ (q -> r) ^ p] -> r true any time r is false. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating: 3 ********************************************* Question: `q006. Explain how we have shown in the past few exercises that [ (p -> q) ^ (q -> r) ^ p] -> r must always be true. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The only difference I’ve noticed is that when it was missing “r” on the outside, it was false, so I would just have to look at it and see whether the argument had “r” in it or not to determine if it was false or true. confidence rating #$&*: 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: We just finished showing that if r is false, [ (p -> q) ^ (q -> r) ^ p] is false so [ (p -> q) ^ (q -> r) ^ p] -> r is true. As seen earlier the statement must also be true whenever r is true. So it's always true. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating: 3 ********************************************* Question: `q007. Explain how this shows that the original argument about rain, wet grass and smelling wet grass, must be valid. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I would just assume that the statement would be valid, because if it rains, the grass becomes wet, and you’d be able to smell wet grass. confidence rating #$&*: 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: That argument is symbolized by the statement [ (p -> q) ^ (q -> r) ^ p] -> r. The statement is always true. There is never a case where the statement is false. Therefore the argument is valid. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating: 3 ********************************************* Question: `q008. Explain how the conclusion of the last example, that [ (p -> q) ^ (q -> r) ^ p] -> r is always a true statement, shows that the following argument is valid: 'If it snows, the roads are slippery. If the roads are slippery they'll be safer to drive on. It just snowed. Therefore the roads are safer to drive on.' Hint: First symbolize the present argument. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: It seems like the statement “If it snows, the roads are slippery. If the roads are slippery they’ll be safer to drive on” would be an invalid argument, because slippery roads are not safer to drive on, but because of this statement, and because what it said would happen does happen, the statement would be valid, although I’m not sure how it could be plugged into the other statement. confidence rating #$&*: 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: This argument can be symbolized by letting p stand for 'it snows', q for 'the roads are slippery', r for 'the roads are safer to drive on'. Then 'If it snows, the roads are slippery' is symbolized by p -> q. 'If the roads are slippery they'll be safer to drive on' is symbolized by q -> r. 'It just snowed' is symbolized by p. 'The roads are safer to drive on' is symbolized by r. The argument the says that IF [ p -> q, AND q -> r, AND p ] are all true, THEN r is true. In symbolic form this is [ (p -> q) ^ (q -> r) ^ p] -> r. This is the same statement as before, which we have shown to be always true. Therefore the argument is valid. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating: 3 ********************************************* 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.' YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I would assume that the statement is valid, because it says that “There is no picnic”, and “If it doesn’t rain, there is a picnic”, so because there is no picnic, you could assume that it rained. The statement could be seen as not valid, because the statement claims that if it does not rain, there will be a picnic. There is not a picnic, but you could figure that it could be for other reasons, and not for rain. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: We could let p stand for 'it rained', q for 'there is a picnic'. The first statement is 'If it doesn't rain there is a picnic', which is symbolized by ~p -> q. The second statement, 'There is no picnic', is symbolized by ~q. The conclusion, 'it rained', is symbolized by p. The argument therefore says IF [ (~p -> q) AND ~q ], THEN p. This is symbolized by [ (~p -> q) ^ ~q ] -> p. We set up a truth table for this argument: p q ~p ~q ~p -> q (~p -> q) ^ ~q [ (~p -> q) ^ ~q ] -> p T T F F T F T T F F T T T T F T T F T F T F F T T F F T ********************************************* Question: `q010. Symbolize the following argument: If it the sun shines, then we'll have a picnic. The sun doesn't shine. Therefore we don't have a picnic. Then set up a truth table to test the validity of the argument. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I’m not sure how to set up a truth table, but the statement would seem valid because the statement claims that if the sun shines, there will be a picnic. The sun doesn’t shine, so you could assume that they didn’t have a picnic for this reason. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ------------------------------------------------ Self-critique Rating: I’ve been frustrated trying to figure out the truth tables and everything, and just am not feeling like I’m understanding it any better. I’m just not understanding how anything can be determined as true or false without being able to plug in a number to the argument. I’m also not understanding how one statement could always be true or false. I’m not sure how to really gain a better understanding of this, because I have read through the information in the textbook several times, and have really tried to work out the problems, but have not been able to come to an answer that seems anywhere near what is right. " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: I’ve been frustrated trying to figure out the truth tables and everything, and just am not feeling like I’m understanding it any better. I’m just not understanding how anything can be determined as true or false without being able to plug in a number to the argument. I’m also not understanding how one statement could always be true or false. I’m not sure how to really gain a better understanding of this, because I have read through the information in the textbook several times, and have really tried to work out the problems, but have not been able to come to an answer that seems anywhere near what is right. " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!