course }߿assignment #015
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19:35:52 `q001. There are 6 questions in this set. The proposition p -> q is true unless p is true and q is false. Construct the truth table for this proposition.
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RESPONSE --> P Q p --> q ______________________ F F T T F F T T T F F T confidence assessment: 2
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19:36:23 The proposition will be true in every case except the one where p is true and q is false, which is the TF case. The truth table therefore reads as follows: p q p -> q T T T T F F F T T F F T
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RESPONSE --> This is the answer that I provided in my work. self critique assessment: 3
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19:39:54 `q002. Reason out, then construct a truth table for the proposition ~p -> q.
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RESPONSE --> p q ~ p -> q ___________________________ T T F T F F T F T F F F F T T T confidence assessment: 2
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19:42:00 This proposition will be false in the T -> F case where ~p is true and q is false. Since ~p is true, p must be false so this must be the FT case. The truth table will contain lines for p, q, ~p and ~p -> q. We therefore get p q ~p ~p -> q T T F T since (F -> T) is T T F F T since (F -> F) is T F T F T since (T -> T) is T F F T T since (T -> F) is F
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RESPONSE --> Whe I worked this problem, I failed to do the ~p step. I almost did, but when I went to work it and set up the truth table, I decided that it wasnt needed for this problem whcih was wrong. Since I left out that step, I was unable actual ""answer"" because I had not completed all the necessary steps. self critique assessment: 2
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19:56:27 `q003. Reason out the truth value of the proposition (p ^ ~q) U (~p -> ~q ) in the case FT (i.e., p false, q true).
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RESPONSE --> (p ^ ~q) U (~p -> ~q ) (F^ ~T) U (~F -> ~T) (F^F) U (T -> F) F U T T confidence assessment: 2
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19:58:24 To evaluate the expression we must first evaluate p ^ ~q and ~p -> ~q. p ^ ~q is evaluated by first determining the values of p and ~q. If p is false and q true, then ~q is false. Thus both p and ~q are false, and p ^ ~q is false. ~p -> ~q will be false if ~p is true and ~q is false; otherwise it will be true. In the FT case p is false to ~p is true, and q is true so ~q is false. Thus it is indeed the case the ~p -> ~q is false. (p ^ ~q) U (~p -> ~q ) will be false if (p ^ ~q) and (~p -> ~q ) are both false, and will otherwise be true. In the case of the FT truth values we have seen that both (p ^ ~q) and (~p -> ~q ) are false, so that (p ^ ~q) U (~p -> ~q ) is false.
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RESPONSE --> when I worked this problem, I was mistaken in my thinking that the TF was ultimately true statement. If I had known that it was False, then the other I got was also false and the final answer in the expression would have come out to be false in my answer - which is the actual correct answer. self critique assessment: 2
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20:00:17 `q004. Construct a truth table for the proposition (p ^ ~q) U (~p -> ~q ).
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RESPONSE --> (p ^ ~q) U (~p -> ~q ) (T ^ ~F) U (~T -> ~F) (T ^ T) U (F -> T) T T True confidence assessment: 2
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20:02:02 We will need headings for p, q, ~p, ~q, (p ^ ~q), (~p -> ~q ) and (p ^ ~q) U (~p -> ~q ). So we set up our truth table p q ~p ~q (p ^ ~q) (~p -> ~q ) (p ^ ~q) U (~p -> ~q ) T T F F F T T T F F T F T T F T T F F F F F F T T F T T To see the first line, where p and q are both T, we first see that ~p and ~q must both be false. (p ^ ~q) will therefore be false, since ~q is false; (~p -> ~q) is of the form F -> F and is therefore true. Since (~p -> ~q) is true, (p ^ ~q) U (~p -> ~q ) must be true. To see the second line, where p is T and q is F, we for see that ~p will be F and ~q true. (p ^ ~q) will therefore be true, since both p and ~q are true; (~p -> ~q) is of the form F -> T and is therefore true. Since (p ^ ~q) and (~p -> ~q ) are both true, (p ^ ~q) U (~p -> ~q ) is certainly true. To see the fourth line, where p is F and q is F, we for see that ~p will be T and ~q true. (p ^ ~q) will be false, since p is false; (~p -> ~q) is of the form T -> T and is therefore true. Since (~p -> ~q ) is true, (p ^ ~q) U (~p -> ~q ) is true.
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RESPONSE --> Apparently I was confused on this question as to the correct way to evaluate this problem. I dont really understand how it gets worked this way (the correct way). self critique assessment: 1
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20:05:34 `q005. If we have a compound sentence consisting of three statements, e.g., p, q and r, then what possible combinations of truth values can occur?
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RESPONSE --> P Q R __________ T T T T F F F F F F T T F T F T T F T F T confidence assessment: 1
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20:06:01 A compound statement with two statements p and q has four possible combinations of truth values: TT, TF, FT, FF. Here we also have r, which can be either T or F. So we can append either T or F to each of the possible combinations for p and q. If r is true then we have possible combinations TT T, TF T, FT T, FF T. If r is false we have TT F, TF F, FT F, FF F. This gives us 8 possible combinations: TTT, TFT, FTT, FFT, TTF, TFF, FTF, FFF.
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RESPONSE --> This is understood and shown in my answer self critique assessment: 3
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20:06:06 A compound statement with two statements p and q has four possible combinations of truth values: TT, TF, FT, FF. Here we also have r, which can be either T or F. So we can append either T or F to each of the possible combinations for p and q. If r is true then we have possible combinations TT T, TF T, FT T, FF T. If r is false we have TT F, TF F, FT F, FF F. This gives us 8 possible combinations: TTT, TFT, FTT, FFT, TTF, TFF, FTF, FFF.
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RESPONSE --> self critique assessment:
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20:08:01 `q006. Evaluate the TFT, FFT and FTF lines of the truth table for (p ^ ~q) -> r.
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RESPONSE --> (p ^ ~q) -> r (T ^ ~F) -> F (F ^ T) -> F F -> F F confidence assessment: 2
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20:08:11 We would need column headings p, q, r, ~q, (p^~q) and (p^~q) -> r. The truth table would then read
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RESPONSE --> self critique assessment: 2
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20:09:14 p q r ~q (p^~q) (p^~q) -> r T F T T T T F F T T F T F T F F F T
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RESPONSE --> Once again, as before I used the wrong method of evaluating this problem. This type of problem is not making sense to me. self critique assessment: 1
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Rmb{x assignment #016 016. Translating Arguments Liberal Arts Mathematics I 07-08-2007
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20:12:02 `q001. There are 6 questions in this set. Is the following argument valid? 'If it rains, the grass will get wet. If the grass gets wet, we'll be able to smell the wet grass. It rained yesterday. Therefore yesterday we were able to smell the wet grass.'
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RESPONSE --> Yes, I believe this statement to be valid. It stands to reason that the statement is correct. confidence assessment: 3
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20:12:20 This argument certainly seems valid. We say what will happen if rains, and what will happen is that happens. Then we say that it rains, so the whole chain of happenings, rained then wet grass then smell, should follow.
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RESPONSE --> self critique assessment: 3
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20:13:39 `q002. Is the following argument valid: 'If it snows, the roads will be slippery. If the roads are slippery they'll be safer to drive on. Yesterday it snowed. Therefore yesterday the roads were safer to drive on.'
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RESPONSE --> This argument does not seem valid because logic would say that one of the statements (if the roads are slippery they'll be safer to drive on) is false. I believe this make the whole statement invalid. confidence assessment: 3
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20:14:41 The validity of an argument has nothing to do with whether the statements in that argument are true or not. All we are allowed to do is assume that the statements are indeed true, and see if the conclusions of the argument therefore hold. In this case, we might well question the statement 'if the roads are slippery they'll be safer to drive on', which certainly seems untrue. However that has nothing to do with the validity of the argument itself. We can later choose to reject the conclusion because it is based on a faulty assumption, but we cannot say that the argument is invalid because of a faulty assumption. This argument tells us that something will happen if it snows, and then tells us what we can conclude from that. It then tells us that it snows, and everything follows logically along a transitive chain, starting from from the first thing.
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RESPONSE --> In this problem, I did exactly what this is telling us not to do. I read too much ""logic"" into the problem, which we are not permitted to do. self critique assessment: 2
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20:15:41 `q003. Is the following argument valid: 'Today it will rain or it will snow. Today it didn't rain. Therefore today it snowed.'
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RESPONSE --> Yes, I believe it is a valid statement. It's telling us that one or the other will happen. It also states that one didnt happen, there the other did happen. confidence assessment: 3
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20:15:57 If we accept the fact that it will do one thing or another, then at least one of those things must happen. If it is known that if one of those things fails to happen, then, the other must. Therefore this argument is valid.
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RESPONSE --> this was stated in my answer self critique assessment: 3
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20:16:45 `q004. Is the following argument valid: 'If it doesn't rain we'll have a picnic. We don't have a picnic. Therefore it rained.'
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RESPONSE --> yes, it is valid. It states that if it doesnt rain, they will have a picnic. They did not have a picnic, therefore it must have rained. confidence assessment: 3
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20:17:06 In this argument where told the something must happen as a result of a certain condition. That thing is not happen, so the condition cannot have been satisfied. The condition was that it doesn't rain; since this condition cannot have been satisfied that it must have rained. The argument is valid.
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RESPONSE --> understood and this answer was given in my answer. self critique assessment: 3
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20:18:42 `q005. We can symbolize the following argument: 'If it rains, the grass gets wet. If the grass gets wet, we'll be able to smell the wet grass. It rained yesterday. Therefore yesterday we were able to smell the wet grass.' Let p stand for 'It rains', q for 'the grass gets wet' and r for 'we can smell the wet grass'. Then the first sentence forms a compound statement which we symbolize as p -> q. Symbolize the remaining statements in the argument.
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RESPONSE --> p -> q U r ->q confidence assessment: 2
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20:19:42 The argument gives three conditions, 'If it rains, the grass gets wet. If the grass gets wet, we'll be able to smell the wet grass. It rained yesterday.', which are symbolized p -> q, q -> r and p. It says that under these three conditions, the statement r, 'we can smell the wet grass', must be true. Therefore the argument can be symbolized by the complex statement [ (p -> q) ^ (q -> r) ^ p] -> r.
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RESPONSE --> I believe I was on the right track with this problem in stating my answer, but I left out some of the steps. This problem did trip me up because I was unable to put all the sentence together. self critique assessment: 2
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20:22:05 `q006. The preceding argument was symbolized as [ (p -> q) ^ (q -> r) ^ p] -> r. Determine whether this statement is true for p, q, r truth values F F T.
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RESPONSE --> [ (p -> q) ^ (q -> r) ^ p] -> r [ (F-> F) ^ (F ->T) ^ F] -> T F ^ T F T confidence assessment: 1
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20:23:05 For these truth values p -> q is true since p is false (recall that the only way p -> q can be false is for p to be true and q to be false), q -> r is false since q is false, and p itself is false, therefore [ (p -> q) ^ (q -> r) ^ p] is false. This makes [ (p -> q) ^ (q -> r) ^ p] -> r true, since the statement can only be false if [ (p -> q) ^ (q -> r) ^ p] is true while r is false.
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RESPONSE --> I think I was on my way to the correct answer, but I failed to work the problem out completely. I got ""lost"" on what to do next at one point, so I thought I had gone as far as the problem required. self critique assessment: 2
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