course MTH 151 mLUdyQѰթassignment #003
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16:37:31 Query 2.3.15 (Y ^ Z')U X, univ={a,..g}, X={a,c,e,g}, Y = {a,b,c}, Z = {b, ..., f} What is the set (Y ^ Z')U X?
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RESPONSE --> ( Y ^ Z' )U X={a,c,e,g} because a,g are not in Z but are in Y and c,e are in X which is in the Union. confidence assessment: 2
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16:37:41 **Z' = {a,g}, the set of all elements of the universal set not in Z. Y ^ Z' = {a}, since a is the only element common to both Y and Z'. So (Y ^ Z') U X = {a, c, e, g}, the set of all elements which lie in at least one of the sets (Y ^ Z') U X. **
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RESPONSE --> Correct self critique assessment: 3
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16:39:35 Give the intersection of the two sets Y and Z'
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RESPONSE --> Y ^ Z'={a} because a is in Y and not in Z. confidence assessment: 1
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16:39:51 **Z' = {a,g}, the set of all elements of the universal set not in Z. Y ^ Z' = {a}, since a is the only element common to both Y and Z'.**
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RESPONSE --> Correct self critique assessment: 3
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16:41:59 Query 2.3.30 describe in words (A ^ B' ) U (B ^ A')
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RESPONSE --> Elements that are in A but are not in B and elements that are in B but not in A. confidence assessment: 1
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16:42:20 ** a description, not using a lot of set-theoretic terms, of (A ^ B' ) U (B ^ A') would be, all the elements that are in A and not in B, or that are not in A and are in B Or you might want to say something like 'elements which are in A but not B OR which are in B but not A'. STUDENT SOLUTION WITH INSTRUCTOR COMMENT:everything that is in set A and not in set B or everything that is in set B and is not in set A. INSTRUCTOR COMMENT: I'd avoid the use of 'everything' unless the word is necessary to the description. Otherwise it's likely to be misleading. **
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RESPONSE --> Correct self critique assessment: 3
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16:44:25 2.3.51 always or not always true: n(A U B) = n(A)+n(B)
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RESPONSE --> I don't think it would always be true because it would depend on what is in set A and B. confidence assessment: 1
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16:44:47 ** This conclusion is contradicted by many examples, including the one of the dark-haired and bright-eyed people in the q_a_. Basically n(A U B) isn't equal to n(A) + n(B) if there are some elements which are in both sets--i.e., in the intersection. } MORE DETAIL: The statement can be either true or false, depending on the sets A and B; it is not always true. The statement n(A U B) = n(A)+n(B) means that the number of elements in A U B is equal to the sum of the number of elements in A and the number of elements in B. The statement would be true for A = { c, f } and B = { a, g, h} because A U B would be { a, c, f, g, h} so n(A U B) = 5, and n(A) + n(B) = 2 + 3 = 5. The statement would not be true for A = { c, f, g } and B = { a, g, h} because A U B would be the same as before so n(AUB) = 5, while n(A) + n(B) = 3 + 3 = 6. The precise condition for which the statement is true is that A and B have nothing in common. In that case n(A U B) = n(A) + n(B). A more precise mathematical way to state this is to say that n(A U B) = n(A) + n(B) if and only if the intersection A ^ B of the two sets is empty. **
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RESPONSE --> Correct self critique assessment: 3
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16:49:35 Query 2.3.60 X = {1,3,5}, Y = {1,2,3}. Find (X ^ Y)' and X' U Y'.
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RESPONSE --> (X ^ Y)' ={2,5} because {1,3} are in both sets and {2,5} are not in both. X' U Y' ={2,5} because again they are not included in both sets just one set. confidence assessment: 1
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16:51:21 ** X ^ Y = {1,3} so (X ^ Y) ' = {1,3}' = {2, 4, 5}. (X ' U Y ' ) = {2, 4} U {4, 5} = {2, 4, 5} The two resulting sets are equal so a reasonable conjecture would be that (X ^ Y)' = X' U Y'. **
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RESPONSE --> I got this question partially correct. but I left out the 2 for some reason. self critique assessment: 2
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16:57:26 2.3.72 A = {3,6,9,12}, B = {6,8}.
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RESPONSE --> A X B ={(3,6),(3,8),(6,6),(6,8),(9,6),(9,8),(12,6),(12,8)} B X A ={(6,3),(6,6),(6,9),(6,12),(8,3),(8,6),(8,9),(8,12)} confidence assessment: 2
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16:57:51 ** (A X B) = {(3,6),(3,8),(6,6),(6,8),(9,6),(9,8),(12,6), (12,8)} (B X A) = (6,3),(6,6),(6,9),(6,12),(8,3),(8,6),(8,9),(8,12)} How is n(A x B) related to n(A) and n(B)? n(S) stands for the number of elements in the set S, i.e., its cardinality. n(A x B) = n(A) * n(B) **
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RESPONSE --> Correct self critique assessment: 3
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16:58:58 2.3.84 Shade A U B
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RESPONSE --> The only thing that would be shaded in would be everything in A and B. confidence assessment: 1
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16:59:07 ** everything in A and everything in B would be shaded. The rest of the universal set (the region outside A and B but still in the rectangle) wouldn't be. **
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RESPONSE --> Correct self critique assessment: 3
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17:05:13 Query 2.3.100 Shade (A' ^ B) ^ C
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RESPONSE --> I think it would be shaded outside the A because it is A' which is not in A and then inside B and C. confidence assessment: 1
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17:05:27 ** you would have to shade every region that lies outside of A and also inside B and also inside C. This would be the single region in the overlap of B and C but not including any part of A. Another way to put it: the region common to B and C, but not including any of A **
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RESPONSE --> Correct self critique assessment: 3
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17:06:34 Describe the shading of the set (A ^ B)' U C.
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RESPONSE --> I think it would be shaded outside of A and B and inside of C. confidence assessment: 1
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17:07:04 ** All of C would be shaded because we have a union with C, which will include all of C. Every region outside A ^ B would also be shaded. A ^ B is the 'overlap' region where A and B meet, and only this 'overlap' would not be part of (A ^ B) '. The 'large' parts of A and B, as well as everything outside of A and B, would therefore be shaded. Combining this with the shading of C the only the part of the diagram not shaded would be that part of the 'overlap' of A and B which is not part of C. **
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RESPONSE --> Correct-I think.
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17:09:13 2.3.114 Largest area of A shaded (sets A,B,C). Write a description using A, B, C, subset, union, intersection symbols, ', - for the shaded region.
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RESPONSE --> I think it would be A^(B' U C') but I am not sure. confidence assessment: 0
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17:10:35 ** Student Answer and Instructor Response: (B'^C')^A Instructor Response: Good. Another alternative would be A - (B U C ), and others are mentioned below. COMMON ERROR: A ^ (B' U C') INSTRUCTOR COMMENT: This is close but A ^ (B' U C') would contain all of B ^ C, including a part that's not shaded. A ^ (B U C)' would be one correct answer. **
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RESPONSE --> I did not get this correct, but I understand how this included a part that is not shaded. I should have put the B without the ' sign. self critique assessment: 2
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{֒\jw assignment #004 004. `Query 4 College Algebra 01-24-2009
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17:23:00 2.4.12 n(A') = 25, n(B) = 28, n(A' U B') = 40, n(A ^ B) = 10
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RESPONSE --> When I made my circles, I put A and B. Inside A I put 15 and in B I put 18 and 10 in the intersection. 25-10=15 and 28-10=18. confidence assessment: 0
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17:24:28 ** In terms of the picture (2 circles, linked, representing the two sets) there are 28 in B and 10 in A ^ B so there are 18 in the region of B outside of A--this is the region B-A. There are 25 outside of A, and 18 of these are accounted for in this region of B. Everything else outside of A must therefore also be outside of B, so there are 25-18=7 elements in the region outside of both A and B. A ' U B ' consists of everything that is either outside of A or outside of B, or both. The only region that's not part of A ' U B ' is therefore the intersection A ^ B, since everything in this region is inside both sets. A' U B' is therefore everything but the region A ^ B which is common to both A and B. This includes the 18 elements in B that aren't in A and the 7 outside both A and B. This leaves 40 - 18 - 7 = 15 in the region of A that doesn't include any of B. This region is the region A - B you are looking for. **
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RESPONSE --> I got this partially correct. I left out part of the answer but I understand that 7 should have been outside both circles. self critique assessment: 2
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17:26:38 query 2.4.18 wrote and produced 2, wrote 5, produced 7 &&&& How many did he write but not produce?
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RESPONSE --> He wrote 5 then subtract 2 for the ones he wrote and produced which would equal 3 that he wrote but did not produce. confidence assessment: 2
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17:27:11 ** You need to count the two he wrote and produced among those he wrote, and also among those he produced. He only wrote 5, two of which he also produced. So he wrote only 3 without producing them. In terms of the circles you might have a set A with 5 elements (representing what he wrote), B with 7 elements (representing what he produced) and A ^ B with 2 elements. This leaves 3 elements in the single region A - B and 5 elements in the single region B - A. The 3 elements in B - A would be the answer to the question. **
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RESPONSE --> Correct self critique assessment: 3
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17:39:31 2.4.24 9 fat red r, 18 thn brown r, 2 fat red h, 6 thin red r, 26 fat r, 5 thin red h, 37 fat, 7 thin brown ......!!!!!!!!...................................
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RESPONSE --> a. 37 b. 9+2+6+5=22 c. 9+26+18+6=59 d. 37-26=11 e. 18+7=25 f. 9+2=11 confidence assessment: 2
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17:41:09 ** Here's my solution. Tell me if there is anything you disagree with (I'm not infallible) or don't understand. incidental: 18 thin brown roosters, 7 thin brown hens, 6 thin red hens and the 6 thin roosters which aren't fat (out of the 50-26=24 thin roosters 18 are brown so 6 are red) adds up to 37 thin chickens How many chickens are fat? 37 as given How many chickens are red? 22: 9 fat red roosters, 6 thin red roosters, 5 thin red hens, 2 fat red hens. How many chickens are male? 50: 9 fat red roosters are counted among the 26 fat roosters so the remaining 17 fat roosters are brown; then there are 18 thin brown roosters and 6 thin red roosters; the number of roosters therefore adds up to 9 + 18 + 6 + 17 = 50 How many chickens are fat not male? 26 of the 37 fat chickens are male, leaving 11 female How many chickens are brown not fat? 25: 18 thin brown roosters, 7 thin brown hens adds up to 25 thin brown chickens How many chickens are red and fat? 11: 9 fat red roosters and 2 fat red hens.**
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RESPONSE --> I got all but 1 correct. I missed c. because I did not subtract 9 from 26 because it was already counted. self critique assessment: 2
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