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23:27:18 `q001. Note that there are 4 questions in this assignment. Suppose I tell you 'If it rains today, I'll give you $100.' Under which of the following circumstances can you claim that I was not telling the truth? 1. It rains and I give you $100. 2. It rains and I don't give you $100. 3. It doesn't rain and I give you $100. 4. I doesn't rain and I don't give you $100.
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RESPONSE --> I can claim you weren't telling the truth with the second statement because it rains and you haven't given me $100. confidence assessment: 1
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23:28:26 `q002. Suppose that tell you 'It will rain today and I will give you $100'. Under which of the following circumstances can you claim that I was not telling the truth? 1. It rains and I give you $100. 2. It rains and I don't give you $100. 3. It doesn't rain and I give you $100. 4. I doesn't rain and I don't give you $100.
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RESPONSE --> The second statement will mean you are not telling the truth. confidence assessment: 1
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23:28:50 It should be clear that situation #1 completely fulfills the conditions of my statement. Both of the things that I say will happen do happen. In situation #2, it rains but you don't get the $100. I said two things were going to happen and one of them didn't. In that case you would have to say that I wasn't telling truth. In situation #3, again one of the things I say is going to happen does but the other doesn't, so again you would have to say that I wasn't telling truth. In situation #4, neither of the things I say will happen does and certainly it would have to be said that I wasn't telling truth.
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RESPONSE --> I realize about the third situation and completely overlooked it. self critique assessment: 1
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23:30:12 `q003. Suppose that tell you 'It will rain today or I will give you $100, but not both'. Under which of the following circumstances can you claim that I was not telling the truth? 1. It rains and I give you $100. 2. It rains and I don't give you $100. 3. It doesn't rain and I give you $100. 4. I doesn't rain and I don't give you $100.
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RESPONSE --> Situation number one I can claim that you weren't telling the truth. Situation four as well because it must rain or you must give me $100 and it doesn't do either. confidence assessment: 1
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23:30:37 `q004. Suppose that tell you 'It will rain today or I will give you $100'. Under which of the following circumstances can you claim that I was not telling the truth? 1. It rains and I give you $100. 2. It rains and I don't give you $100. 3. It doesn't rain and I give you $100. 4. I doesn't rain and I don't give you $100.
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RESPONSE --> I understand this concept a little better now. confidence assessment: 1
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????????n?????assignment #012 012. The common sense of logic. Liberal Arts Mathematics I 04-04-2009 ???????????? assignment #013 013. Negation Liberal Arts Mathematics I 04-04-2009
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23:33:52 `q001. There are 4 questions in this set. Two statements are said to be negations of one another if exactly one of the statements must be true. This means that if one statement is true the other must be false, and if one statement is false the other must be true. What statement is the negation of the statement 'all men are over six feet tall'?
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RESPONSE --> Some men are over six feet tall. confidence assessment: 1
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23:34:17 You might think that the negation would be 'no men are over six feet tall'. However, the negation is in fact 'some men are not over 6 feet tall'. The negation of a statement, in addition to being false whenever the statement is true, has to include every possibility except those covered by the statement itself. With respect to men being over six feet tall, there are three possibilities: 1. All men are over six feet tall, 2. no men are over six feet tall, and 3. some men are over six feet tall while others aren't. It should be clear that statements 1 and 2 do not cover the possibility of the third. In fact no two of these statements cover the possibility of the remaining one. However the following two statements do cover all possibilities: All men are over six feet tall (the original statement), and some men are not over six feet tall. The second statement might seem to be identical to statement 3, 'some men are over six feet tall while others aren't', but it is not. The statement 'some men are not over six feet tall' does not address whether there are men over six feet tall or not, while statement 3 states that there are. And the statement 'some men are not over six feet tall' might seem to leave out the possibility of statement 2, 'no men are over six feet tall', but again it doesn't address whether or not there are also men over six the tall. Therefore the negation of the statement 'all men are over six feet tall' is 'some men are not over six feet tall'.
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RESPONSE --> self critique assessment: 3
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23:35:20 `q002. What is the negation of the statement 'some men are over six feet tall' ?
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RESPONSE --> No men are over six feet tall. confidence assessment: 1
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23:35:33 While it might seem that the negation of this statement is 'some men are not over six feet tall', the correct negation is 'no men are over six feet tall'. This is because there is an 'overlap' between 'some men are over six feet tall' and 'some men are not over six feet tall' because both statements are true if some men are over six feet while some are under six feet. Negations have to be exact opposites--if one statement is true the other must be false--in addition to the condition that the two statements cover every possible occurance. Again we have the three possibilities, 1. All men are over six feet tall, 2. no men are over six feet tall, and 3. some men are over six feet tall while others aren't. The statement ' some men are over six feet tall' is consistent with statements 1 and 3, because if all men are over six feet tall then certainly some men are over 6 feet tall, and if some men are over 6 feet tall and others aren't, it is certainly true that some men are over six feet tall. The only statement not consistent with 'some men are over six feet tall' is Statement 2, 'No men are over six feet tall'. Thus this statement is the negation we are looking for.
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RESPONSE --> self critique assessment: 3
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23:37:12 `q003. As seen in the preceding two questions, the negation of a statement that says 'all are' or 'all do' is 'some aren't' or 'some don't', and the negation of a statement that says 'some are' or 'some do' is 'all aren't' or 'none are', or 'all do not' or 'none do'. Each of the following statements can be expressed as and 'all' statement or a 'some' statement. Identify which is which and give the negation of each statement: 1. Every dog has its day. 2. Some roses are black. 3. Every attempt fails. 4. In some cases the desired outcome isn't attained.
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RESPONSE --> 1 is an all statement and the negation is some dogs have their day 2 is a some statement and the negation is No roses are black 3 is an all statement and its negation is Some attempts fail 4 is a some statement and its negation is In no cases the desired outcome isn't attained. confidence assessment: 1
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23:37:24 Statement 1 can be expressed as 'All dogs do have their day', a form of 'all do'. The negation of 'all do' is 'some don't'. In this case the negation might be expressed as 'some dogs do not have their day'. Statement 2 is a straightforward 'some are' statement having negation 'all are not', expressed in this case as 'no roses are black', or equivalently 'there are no black roses'. Statement 3 can be restated equivalently in 'all do' form as 'all attempts do fail', and is negated in 'some don't' form as 'some attempts do not fail', or equivalently as 'some attempts succeed'. Statement 4 can be equivalently expressed in 'some are' form as 'some outcomes are not as desired'. This statement is negated by the 'none are' form as 'no outcomes are not as desired', which can then be expressed as 'all outcomes are as desired'.
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RESPONSE --> self critique assessment: 3
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23:38:07 `q004. Negate the following statements: 1. No roses are black. 2. Some roses are not black. 3. There were Dodo birds that weren't stupid. 4. There were never turtles that weren't slow.
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RESPONSE --> Some roses are black. No roses aren't black. All Dodo Birds weren't stupid. All turtles were slow. confidence assessment: 1
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23:40:21 `q001. There are 8 questions in this set. If each of the propositions p and q can be either true or false, what combinations of truth values are possible for the two propositions (e.g., one possibility is that p is false and q is true; list the other possibilities)?
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RESPONSE --> p q T F T T F T F F confidence assessment: 1
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23:40:49 It is possible that p is true and q is true. Another possibility is that p is true and q is false. A third possibility is that p is false and q is true. A fourth possibility is that p is false and q is false. These possibilities can be listed as TT, TF, FT and FF, where it is understood that the first truth value is for p and the second for q.
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RESPONSE --> I should have put this in words instead of a table form. self critique assessment: 1
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23:41:44 `q002. For each of the for possibilities TT, TF, FT and FF, what is the truth value of the compound statement p ^ q ?
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RESPONSE --> For TT, it is true, for TF it is False, for FT it is also false and for FF it is true. This is because in an and statement both of the p and q must be the same. confidence assessment: 1
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23:42:05 p ^ q means 'p and q', which is only true if both p and q are true. In the case TT, p is true and q is true so p ^ q is true. In the case TF, p is true and q is false so p ^ q is false. In the case FT, p is false and q is true so p ^ q is false. In the case FF, p is false and q is false so p ^ q is false.
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RESPONSE --> I messed up on the FF it had me slightly confused. I understand my mistake now. self critique assessment: 1
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23:42:55 The truth table must have headings for p, q and p ^ q. It must include a line for each of the possible combinations of truth values for p and q. The table is as follows: p q p ^ q T T T T F F F T F F F F.
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RESPONSE --> p q p^q T T T T F F F T F F F F self critique assessment: 1
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23:43:55 `q004. For each of the possible combinations TT, TF, FT, FF, what is the truth value of the proposition p ^ ~q?
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RESPONSE --> False, True, False, False. The second, TF, is the only one that matches since it would become TT in a p^ ~q statement. confidence assessment: 1
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23:44:01 For TT we have p true, q true so ~q is false and p ^ ~q is false. For TF we have p true, q false so ~q is true and p ^ ~q is true. For FT we have p false, q true so ~q is false and p ^ ~q is false. For FF we have p false, q false so ~q is true and p ^ ~q is false.
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RESPONSE --> self critique assessment: 3
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23:44:50 The truth table will have to have headings for p, q, ~q and p ^ ~q. We therefore have the following: p q ~q p^~q T T F F T F T T F T F F F F T F
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RESPONSE --> p q ~q p^~q T T F F T F T T F T F F F F T F self critique assessment:
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23:44:55 `q006. Give the truth table for the proposition p U q, where U stands for disjunction.
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RESPONSE --> confidence assessment:
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??j??????????? assignment #014 014. Truth Tables Liberal Arts Mathematics I 04-04-2009
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23:46:18 p U q means 'p or q' and is true whenever at least one of the statements p, q is true. Therefore p U q is true in the cases TT, TF, FT, all of which have at least one 'true', and false in the case FF. The truth table therefore reads p q p U q T T T T F T F T T F F F
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RESPONSE --> p q pUq T T T T F T F T T F F F self critique assessment:
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23:48:12 In the case TT p is true and q is true, so ~q is false. Thus p U ~q is true, since p is true. So ~(p U ~q) is false. In the case TF p is true and q is false, so ~q is true. Thus p U ~q is true, since p is true (as is q). So ~(p U ~q) is false. In the case FT p is false and q is true, so ~q is false. Thus p U ~q is false, since neither p nor ~q is true. So ~(p U ~q) is true. In the case FF p is false and q is false, so ~q is true. Thus p U ~q is true, since ~q is true. So ~(p U ~q) is false.
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RESPONSE --> ~(p U~q) false TT ~(pU~q) false TF ~(pU~q) true FT ~(pU~q) false FF self critique assessment:
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23:49:06 We need headings for p, q, ~q, p U ~q and ~(p U ~q). Our truth table therefore read as follows: p q ~q pU~q ~(pU~q) T T F T F T F T T F F T F F T F F T T F
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RESPONSE --> p q ~q pU~q ~(pU~q) T T F T F T F T T F F T F F T F F T T F self critique assessment: 1
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???????W??}?P? assignment #014 014. Truth Tables Liberal Arts Mathematics I 04-04-2009