course Mth 151 kϱיh蛧yČassignment #011
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18:02:05 `q001. . There are five questions in this set. Draw four points in a square pattern (i.e., if the points were properly connected, they should form a square). From each of the points, draw a straight line to each of the other points. How many lines did you draw?
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RESPONSE --> six lines were drawn. Four the make the ""box"" of the square, and two to make an ""x"" in the center of the box. confidence assessment: 3
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18:03:18 Each corner of the square will connected to each of the other three corners, so from each corner you would have drawn three lines. Since there are four corners, had you followed the instructions precisely you would have drawn 4 * 3 = 12 lines. However each of these lines will be identical with another line you would have drawn, since for any two corners you would be drawing a line from the first to the second then another overlapping line from the second to the first. Therefore you might have said that there are 6 lines.
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RESPONSE --> I did this problem with the statement of drawing 6 lines. self critique assessment: 3
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18:03:58 `q002. How many triangles are there in the figure you drew?
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RESPONSE --> I see four triangles in the figure. One making up each side in the box confidence assessment: 3
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18:05:12 You should have a total of 8 triangles. The diagonals divide the square up into 4 small triangles. Each diagonal also divides the square into 2 larger triangles. Since there are 2 diagonals there are 4 larger triangles. The 4 small triangles and the 4 larger triangles total 8 triangles.
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RESPONSE --> I didnt see the larger triangle images before seeing this answer. After reading this and taking another look, I do see where they got the eight total triangles. self critique assessment: 3
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18:07:17 `q003. The lines you drew should form a square with its two diagonals. Label the corners of the square A, B, C and D, going in order around the square, and label the center where the diagonals cross E. Now list all possible combinations of 3 of the letters A, B, C, D, E (note: combinations don't care about order, so A D E is the same as D A E or E A D or any other combination of these same three letters, so list each possible combination only once. That is, if you list for example ADE you won't list DAE).
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RESPONSE --> ABC AEC DEB BCD DAB I got five combinations confidence assessment: 3
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18:08:53 The easiest way to list these sequences is alphabetically: ABC, ABD, ABE all start with AB; then ACD and ACE start with AC and ADE starts with AD. This is a list of all possible combinations containing A. We next list all possible remaining combinations containing B: BCD, BCE and BDE. Then we write down CDE, the only remaining combination containing C. We thus have the 10 combinations ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE.
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RESPONSE --> Apparently I failed to look at ALL possibilities. For some reason I just looked at all the ""A"" combinations, and came up with five. I think maybe I rushed through it before giving it enough time to think through it. It's an easy answer to see after looking ahead and getting the answers. self critique assessment: 3
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18:10:16 `q004. Of the 10 combinations ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE, which form triangles on your figure?
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RESPONSE --> ABD AED DEC BEC BDE ABD All of these form triangles. confidence assessment: 3
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18:11:38 ABC forms a large triangle from A to B to C and back to A. The same is true of ABD, ACD and BCD. These are the large triangles in the figure. ACE and BDE form straight lines, not triangles. ABE, ADE, BCE and CDE form small triangles. Thus of the 10 possible combinations of labeled points, we find the 4 large triangles and the 4 small triangles we saw earlier, in addition to 2 straight lines which do not for triangles. Since any triangle in the figure must be labeled by three of the five points A, B, C, D, E, we see that these are the only triangles that can be formed.
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RESPONSE --> I am missing four of the answers they got. Looking back at my drawing, I can still only see the ones I picked out. I'm not sure which ones I'm missing. self critique assessment: 2
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18:13:01 `q005. Sketch the same figure as before, but without the line segment from A to B. Now how may triangles are there?
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RESPONSE --> I count three triangles confidence assessment: 2
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18:13:56 Any possible triangle must still come from the list ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE. We again see that ACE and BDE form straight lines so do not count as triangles. Now ABC, ABD and ABE do not form triangles because the line segment AB is now missing. This leaves us the five triangles ACD, ADE, BCE, BCE and CDE.
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RESPONSE --> I see how they got this answer. Once again, I failed to see to two larger trianglesBCD and ADC self critique assessment: 2
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Mz~QǩEᙫ̅a assignment #012 012. The common sense of logic. Liberal Arts Mathematics I 07-04-2007
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18:16:37 `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 --> It rains and I don't give you $100. I would technically think this is the only one. Another option might be number 3, but with that one, someone could argue that you never said you wouldnt give $100 if it didnt rain confidence assessment: 2
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18:17:32 I said what would happen under a certain condition. In situation #2, that condition is fulfilled and what I said would happen doesn't happen. Therefore in situation #2 it is clear that I wasn't telling the truth. In situation #3, the condition that I addressed isn't fulfilled so no matter what happens I can't be accused of not telling the truth. I said what would happen if rains. No matter what happens, if it doesn't rain what I said cannot be held against me. It should be clear to anybody that situation #1 is exactly what you would expect, and that situation #4 is just would you would probably expect from my statement in the event that it doesn't rain, so nobody would say that this situation violates my claim to truthfulness.
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RESPONSE --> This is understood, so no critique is needed self critique assessment: 3
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18:18:40 `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 --> 3. It doesn't rain and I give you $100 4. I doesn't rain and I don't give you $100. confidence assessment: 3
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18:19:46 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 left out one of the answers needed. I believe I simply read the statement wrong. I did have to go back and check myself when I first started, because of confusion on my part. self critique assessment: 2
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18:20:20 `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 --> 4. I doesn't rain and I don't give you $100. confidence assessment: 3
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18:20:32 In situations 2 and 3, one of the things happens and the other doesn't, so you would not be able to say that I wasn't telling the truth. However in situation 1, both things happen, which I said wouldn't be the case; and in situation 4 neither thing happens. In both of these situations you would have to say that I was not telling truth.
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RESPONSE --> self critique assessment: 3
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18:21:14 `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 --> 3. It doesn't rain and I give you $100. 4. I doesn't rain and I don't give you $100. confidence assessment: 3
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18:21:56 At first this might seem to be the same as the preceding problem. But in the preceding problem we specifically said '... but not both.' In this case that qualification was not made. Therefore we have regard the statement as true as long as at least one of the conditions is fulfilled. This is certainly the case for situation 1: both conditions are true we can certainly say that at least one is true. So in situation #1 we have to regard the present statement as true. So situation #1 would not be included among those in which I could be accused of not telling the truth.
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RESPONSE --> This is understood, and what I was trying to get at with my answers. self critique assessment: 2
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|nȄƀ assignment #013 013. Negation Liberal Arts Mathematics I 07-04-2007
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18:23:01 `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 --> all over six feet tall are men confidence assessment: 2
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18:24:06 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 --> I was confused by the instructions, but the answer is clear self critique assessment: 2
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18:24:32 It doesn't matter what's true and what isn't. If the question was to write the negation of 'all men are under 20 feet tall' you would still state the negation as 'some men are under 20 feet tall'. In this case the negation is true, which proves that the statement itself is false.
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RESPONSE --> understood self critique assessment: 3
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18:24:43 In the given problem the negation 'some men are under 6 ft tall' is true, proving that the original statement 'all men are over 6 ft tall' is false.
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RESPONSE --> self critique assessment: 3
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18:25:00 These examples demonstrate why it is important to figure out the negation before you even thing about which statement is true. Either the statement or its negation will be true, but never both.
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RESPONSE --> understood self critique assessment: 3
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18:25:29 `q002. What is the negation of the statement 'some men are over six feet tall' ?
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RESPONSE --> all men are over six feet tall confidence assessment: 3
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18:26:15 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 --> understood self critique assessment: 3
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18:29:17 `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. Every dog has its day (all) some dogs dont have their day 2. Some roses are black. (some) all roses aret black 3. Every attempt fails. (all) Some attempts dont fail 4. In some cases the desired outcome isn't attained. (some) In all cases the desired outcome is attained. confidence assessment: 2
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18:30:48 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 --> As far as these statements are concerned, I got some correct and some wrong. I think over all, I am not putting enough thought into the tasks, and this is a personal flaw because I an anxious. After reading this discussion, I can see how they came upon the answers, but it is still a little confusing. self critique assessment: 2
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18:32:00 `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 --> 1. Some roses are not black 2. All roses are black 3. All Dodo birds were stupid 4.Not all turtles were slow confidence assessment: 3
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18:32:38 Statement 1 says that there is no such thing as a rose which is not black, which says that all roses fail to be black. The negation of 'all are' is 'some aren't', so the negation of 'all roses are not black' is 'some roses are not not black', which is the same as 'some roses are black'. Statement 2 is a 'some are' statement, negated in the 'all are not' form by 'all roses are not not black', or equivalently, 'all roses are black'. Statement 3 is equivalent to saying that 'some Dodos birds were not stupid', negated as 'all are not' in the form 'all Dodo birds were not not stupid', or equivalently as 'all Dodo birds were stupid'. Statement 4 is equivalent of saying that 'all turtles were slow', equivalent of the 'all are' form. This is negated in 'some are not' form by 'some turtles were not slow'.
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RESPONSE --> self critique assessment:
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