course Mth 151 ???o??{?v????assignment #001001. Only assignment: prelim asst
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10:38:53 `q001. Part 1 includes six activities. If you have completed an activity, just enter the answer 'completed'. This question is appearing in the Question box. The box to the right is the Answer box, where you will type in your answers to the questions posed here. To use this program you read a question, then enter your answer in the Answer box and click on Enter Answer. In your answers give what is requested, but don't go into excruciating detail. Try to give just enough that the instructor can tell that you understand an item. After entering an answer click on Next Question/Answer above the Question box. Do you understand these instructions?
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RESPONSE -->
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10:38:55 This program has created the folder c:\vhmthphy on your hard drive. Browse to that folder and locate the file whose name begins with SEND. The name of this file will also include your name, as you gave it to the program, and the file will show as a Text file. Never tamper with a SEND file in any way. It contains internal codes as if these codes are tampered with you won't get credit for the assignment. However you are welcome to copy this file to another location and view it, make changes, etc. Just be sure that when requested to do so you send the instructor the original, tamper-free file. State in the Answer box whether or not you have been able to locate the SEND file. Don't send the SEND file yet. Note that more questions/instructions remain in the q_a_prelim.
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10:38:56 `q002. Note that every time you click on Enter Answer the program writes your response to your SEND file. Even if the program disappears all the information you have entered with the Enter Answer button will remain in that file. This program never 'unwrites' anything. Even if this program crashes your information will still be there in the SEND file. Explain this in your own words.
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RESPONSE -->
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10:38:58 Any time you do not receive a reply from the instructor by the end of the following day, you should resubmit your work using the Resubmit Form at http://www.vhcc.edu/dsmith/genInfo/. You have already seen that page, but take another look at that page and be sure you see the Submit Work form, the Resubmit Form and a number of other forms that will be explained later. Enter a sentence or two describing the related links you see at that location.
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10:38:59 `q003. If you are working on a VHCC computer, it is probably set up in such a way as to return to its original configuration when it is rebooted. To avoid losing information it is suggested that you back up your work frequently, either by emailing yourself a copy or by using a key drive or other device. This is a good idea on any computer. Please indicate your understanding of this suggestion.
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RESPONSE -->
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10:39:02 Once more, locate the SEND file in your c:\vhmthphy folder, and open the file. Copy its contents to the clipboard (this is a common operation, but in case you don't know how, just use CTRL-A to highlight the contents of the file and CTRL-C to copy the contents to the clipboard). Then return to the form that instructed you to run this program, and paste the contents into the indicated box (just right-click in the box and select Paste). You may now click on the Quit button, or simply close the program.
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RESPONSE -->
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????????????p?|w Student Name: assignment #006
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10:40:51 `q001. Note that there are 6 questions in this assignment. Find the likely next element of the sequence 1, 2, 4, 7, 11, ... .
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RESPONSE --> 16
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?????????€?Student Name: assignment #006
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10:46:04 `q002. Find the likely next two elements of the sequence 1, 2, 4, 8, 15, 26, ... .
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RESPONSE --> 31
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10:47:11 The difference between 1 and 2 is 1; the difference between 2 and 4 is 2, the difference between 4 and 8 is 4; the difference between 8 and 15 is 7; the difference between 15 and 26 is 11. The differences form the sequence 1, 2, 4, 7, 11, ... . As seen in the preceding problem the differences of this sequence are 1, 2, 3, 4, ... . We would expect the next two differences of this last sequence to be 5 and 6, which would extend the sequence 1, 2, 4, 7, 11, ... to 1, 2, 4, 7, 11, 16, 22, ... . If this is the continuation of the sequence of differences for the original sequence 1, 2, 4, 8, 15, 26, ... then the next two differences of this sequence would be 16 , giving us 26 + 16 = 42 as the next element, and 22, giving us 42 + 26 = 68 as the next element. So the original sequence would continue as 1, 2, 4, 8, 15, 26, 42, 68, ... .
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RESPONSE --> I added wrong. I added 15 to 26, rather than 16.
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10:48:02 `q003. What would be the likely next element in the sequence 1, 2, 4, 8, ... . It is understood that while this sequence starts off the same as that in the preceding exercise, it is not the same. The next element is not 15, and the pattern of the sequence is different than the pattern of the preceding.
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RESPONSE --> 16-multiply by 2
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?????v????F???? Student Name: assignment #007
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10:51:02 `q001. Note that there are 7 questions in this assignment. Sketch three points A, B and C forming an equilateral triangle on a piece of paper, with point A at the lower left-hand corner, point B at the lower right-hand corner and point C at the top. Sketch the segments AB and AC. Now double the lengths of AB and AC, and place a point at each of the endpoints of these segments. Connect these new endpoints to form a new equilateral triangle. Two sides of this triangle will have three points marked while the new side will only have its two endpoints marked. Fix that by marking that middle point, so all three sides of your new triangle are marked the same. How many marked points were there in the original triangle, and how many are there in the new triangle?
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RESPONSE --> there were three and now there are six
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10:53:09 `q002. Extend the two sides that meet at A by distances equal to the distance original lengths AC and AB and mark the endpoints of the newly extended segments. Each of the newly extended sides will have 4 marked points. Now connect the new endpoints to form a new right triangle. Mark points along the new side at the same intervals that occur on the other two sides. How many marked points are on your new triangle, and how many in the whole figure?
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RESPONSE --> there were 6 ad now there are 11.
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10:54:25 You added the two new endpoints when you extended the sides. You then should have marked two new points on the new third side, so that each side contains 4 points including its endpoints. Your figure will now contain 10 marked points.
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RESPONSE --> when i connected the final end points, i put five points on it instead of four, giving me 11. oops
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10:54:58 `q003. Continue the process for another step-extend each side by a distance equal to the original point-to-point distance. How many points do you have in the new triangle?
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RESPONSE --> 14
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10:55:18 You will add an endpoint to each newly extended side, so each of the new sides will contain 5 points. You will then have to add 3 equally spaced points to the new side, giving you a total of 13 points on the new triangle. In addition there are two marked points inside the triangle, for a total of 15 points. Click on 'Next Picture' to see the construction. The line segments along two sides of the triangle have again been extended and points marked at the ends of these segments. The new endpoints have been connected to form the third side of a larger triangle, and equally spaced points have been constructed along that side.
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RESPONSE --> haha now i only added four instead of 5!!
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10:56:38 `q004. Continue the process for one more step. How many points do you have in the new triangle?
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RESPONSE --> 21
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10:56:46 You will add an endpoint to each newly extended side, so each of the new sides will contain 6 points. You will then have to add 4 equally spaced points to the new side, giving you a total of 15 points on the new triangle. There are also 5 marked points inside the triangle for a total of 21 marked points.
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RESPONSE --> i finally got it right
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10:57:12 `q005. The sequence of marked points is 3, 6, 10, 15, 21. What do expect will be the next number in this sequence?
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RESPONSE --> 28 (add 7) one more point being added
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10:57:27 The differences between these numbers are 3, 4, 5, 6. The next difference, according to this pattern, should be 7, which would make the next number 28.
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RESPONSE --> ??
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10:58:00 `q006. How can you tell, in terms of the process you used to construct these triangles, that the next number should be 7 greater?
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RESPONSE --> each time i added a side to the triangle, one more point was added
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10:58:05 When you extend the triangle again, you will add two new endpoints and each side will now have 7 points. The 7 points on the new triangle will be all of the new points.
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RESPONSE -->
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10:58:34 `q007. How do you know this sequence will continue in this manner?
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RESPONSE --> because you can continue to extend the triangle and it will be the same each time
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?????€????????? Student Name: assignment #008
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11:00:52 `q001. There are seven questions in this set. See if you can figure out a strategy for quickly adding the numbers 1 + 2 + 3 + ... + 100, and give your result if you are successful. Don't spend more than a few minutes on your attempt.
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RESPONSE --> 101/50
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11:01:08 These numbers can be paired as follows: 1 with 100, 2 with 99, 3 with 98, etc.. There are 100 number so there are clearly 50 pairs. Each pair adds up to the same thing, 101. So there are 50 pairs each adding up to 101. The resulting sum is therefore total = 50 * 101 = 5050.
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RESPONSE --> i put divided by, instead of multiplied by
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11:01:27 `q002. See if you can use a similar strategy to add up the numbers 1 + 2 + ... + 2000.
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RESPONSE --> 2001*1000
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11:01:34 Pairing 1 with 2000, 2 with 1999, 3 with 1998, etc., and noting that there are 2000 numbers we see that there are 1000 pairs each adding up to 2001. So the sum is 1000 * 2001 = 2,001,000.
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RESPONSE --> ok
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11:02:12 `q003. See if you can devise a strategy to add up the numbers 1 + 2 + ... + 501.
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RESPONSE --> 502*250
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11:02:37 We can pair 1 with 501, 2 with 500, 3 with 499, etc., and each pair will have up to 502. However there are 501 numbers, so not all of the numbers can be paired. The number in the 'middle' will be left out. However it is easy enough to figure out what that number is, since it has to be halfway between 1 and 501. The number must be the average of 1 and 501, or (1 + 501) / 2 = 502 / 2 = 266. Since the other 500 numbers are all paired, we have 250 pairs each adding up to 502, plus 266 left over in the middle. The sum is 250 * 502 + 266 = 125,500 + 266 = 125,751. Note that the 266 is half of 502, so it's half of a pair, and that we could therefore say that we effectively have 250 pairs and 1/2 pair, or 250.5 pairs. 250.5 is half of 501, so we can still calculate the number of pairs by dividing the total number of number, 501, by 2. The total sum is then found by multiplying this number of pairs by the sum 502 of each pair: 250.5 * 502 = 125,766.
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RESPONSE --> ok
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11:03:31 `q004. Use this strategy to add the numbers 1 + 2 + ... + 1533.
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RESPONSE --> 1534*766.5
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11:03:37 Pairing the numbers, 1 with 1533, 2 with 1532, etc., we get pairs which each adult to 1534. There are 1533 numbers so there are 1533 / 2 = 766.5 pairs. We thus have a total of 1534 * 766.5, whatever that multiplies out to (you've got a calculator, and I've only got my unreliable head).
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RESPONSE --> ok
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11:06:20 `q005. Use a similar strategy to add the numbers 55 + 56 + ... + 945.
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RESPONSE --> 945-55=890 890/2=445 945+55=990 445*990
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11:06:29 We can pair up 55 and 945, 56 and 944, etc., obtaining 1000 for each pair. There are 945 - 55 + 1 = 891 numbers in the sum (we have to add 1 because 945 - 55 = 890 tells us how many 1-unit 'jumps' there are between 55 and 945--from 55 to 56, from 56 to 57, etc.. The first 'jump' ends up at 56 and the last 'jump' ends up at 945, so every number except 55 is the end of one of the 890 'jumps'. But 55 is included in the numbers to be summed, so we have 890 + 1 = 891 numbers in the sum). If we have 891 numbers in the sum, we have 891/2 = 445.5 pairs, each adding up to 1000. So we have a total of 445.5 * 1000 = 445,500.
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RESPONSE --> ok
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11:07:42 `q006. Devise a strategy to add the numbers 4 + 8 + 12 + 16 + ... + 900.
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RESPONSE -->
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11:07:59 Pairing 4 with 900, 8 with 896, etc., we get pairs adding up to 904. The difference between 4 and 900 is 896. The numbers 'jump' by 4, so there are 896 / 4 = 224 'jumps'. None of these 'jumps' ends at the first number so there are 224 + 1 = 225 numbers. Thus we have 225 / 2 = 112.5 pairs each adding up to 904, and our total is 112.5 * 904.
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RESPONSE --> ok
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11:10:27 `q007. What expression would stand for the sum 1 + 2 + 3 + ... + n, where n is some whole number?
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RESPONSE --> n(n+1)/2
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11:10:42 We can pair 1 and n, 2 and n-1, 3 and n-2, etc., in each case obtaining a sum of n + 1. There are n numbers so there are n/2 pairs, each totaling n + 1. Thus the total is n/2 * (n+1).
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RESPONSE --> ok
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??????????????i? Student Name: assignment #009
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?yG????????????Student Name: assignment #011
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11:12:05 `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 --> 6
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11:12:22 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 --> ok
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11:12:40 `q002. How many triangles are there in the figure you drew?
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RESPONSE --> 8
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11:12:45 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 --> ok
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11:18:40 `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 --> ABD, BCP, BAC, CAD, AEB, AEC, DEC, DEB, CEA, BDE, BCE, ADE
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11:18:46 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 --> OK
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11:21:00 `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 --> ALL EXCEPT 'AEC' AND 'BDE'
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11:21:16 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 form 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 --> ok
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11:21:43 `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 --> 5
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11:21:53 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 --> ok
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?????????O€y|E??? Student Name: assignment #012
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11:22:41 `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 --> 2
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11:22:47 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 --> ok
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11:23:13 `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 --> 2
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11:23:59 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 didn't read the question thouroughly :(
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11:24:31 `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 --> 1 & 4
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11:24:41 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 --> ok
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11:25:05 `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 --> 4
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11:25:13 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 --> ok
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e???q?????J??e? Student Name: assignment #013
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11:25:59 `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 not over six feet tall
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11:26:31 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 --> once, again, I wasn't reading the question thoroughly
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11:26:45 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 --> ok
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11:26:53 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 --> ok
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11:27:00 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 --> k
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11:27:50 `q002. What is the negation of the statement 'some men are over six feet tall' ?
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RESPONSE --> most men are under six feet tall
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11:28:04 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 --> ok
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11:30:27 `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. all statement negation: some dogs have their day. 2. some statement negation: all roses are black 3. all statement negation: Some attempts fail 4. some statement negation: in all cases, the desired outcome is not attained
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11:30:30 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 --> ok
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11:31:28 `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 all roses are not black there were some dodo birds that were stupid there were never turtles that were slow
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11:31:38 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 --> ok
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?^???K??????N?? Student Name: assignment #014
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11:32:34 `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 is false and q is false p is true and q is false q is true and q is true
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11:32:38 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 --> ok
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11:33:36 `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 --> p^q: tt: true tf: false ft: false ff: false
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11:33:42 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 --> ok
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11:35:07 `q003. Write the results of the preceding problem in the form of a truth table.
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RESPONSE --> p q I p^q ------------------- T T True T F False F T False F F False
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11:35:14 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 --> ok
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11:36:59 `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 --> 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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11:37:03 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 --> OK
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11:37:19 `q005. Give the results of the preceding question in the form of a truth table.
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RESPONSE --> oops.... already did
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11:37:24 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 -->
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11:38:28 `q006. Give the truth table for the proposition p U q, where U stands for disjunction.
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RESPONSE --> p q pUq ------------------ T T T T F T F T T F F F
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11:38:31 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 -->
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11:39:45 `q007. Reason out the truth values of the proposition ~(pU~q).
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RESPONSE --> the q is going to loose its negative property and the p will gain it. So, in three cases, the statement will be true and in one it will be false. It is an 'or' so there can be one false and one true and the statement is still true. (except in the case of two falses.)
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11:39:48 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 --> ok
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11:41:12 `q008. Construct a truth table for the proposition of the preceding question.
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RESPONSE --> p q ~p ~pUq T T F T T F F F F T T T F F T T
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11:41:15 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 --> OK
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???????????Student Name: assignment #015
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11:42:10 `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 T T T T F F F T T F F T
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11:42:14 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 --> OK
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11:43:11 `q002. Reason out, then construct a truth table for the proposition ~p -> q.
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RESPONSE --> p q ~p ~p->q T T F T T F F T F T T T F F T F
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11:43:17 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 --> OK
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