course Mth 151 w}̫assignment #003
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20:17:37 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 --> First you have to find (Y ^ Z') which is {a} Y = {a, b, c} intersect with Z' = {a, g} - you pick which ones are similar in both sets The next step is {a} U X which is {a, c, e, g} You add {a} and all the elements of set X
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20:18:16 **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 --> You say it much better than I do, but I think I have the understanding of it.
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20:20:05 Give the intersection of the two sets Y and Z'
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RESPONSE --> All the elements that are the same in set Y and the complement of set Z. {a, b, c} ^ {a, g} {a}
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20:20:14 **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 --> ok
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20:21:01 Query 2.3.30 describe in words (A ^ B' ) U (B ^ A')
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RESPONSE --> The set of all elements that are in A and not in B, or in B not but in A.
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20:22:59 ** 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 --> ok
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20:25:12 2.3.51 always or not always true: n(A U B) = n(A)+n(B)
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RESPONSE --> Not always true. If I replace the set letters with numbers, this equation does not work. n(A)=7, n(B)=5 n(35) = 12 is not true
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20:29:09 ** 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 --> I messed up on this one. The part of the explanation that made sense to me was... ""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."" I was getting the cardinality and multiplying mixed up. I can't think of where multiplying comes into practice.
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20:37:36 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)' = all elements that are in X intersect with all elements with Y {1, 3} and the complement of this is... {2, 4, 5} X' U Y' = the union of the complements of X and Y {2, 4, 5}
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20:39:44 ** 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 need to write it out like I did on paper rather than trying to say what I am doing. I also didn't write the conjecture. Basically the two sets are equal. (X ^ Y)' = X' U Y'
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20:42:47 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)}
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20:47:00 ** (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 --> ok
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20:49:11 2.3.84 Shade A U B
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RESPONSE --> If I use the numbers that the book uses where U = 1, A = 2, AB = 3 and B = 4, then the only shaded part is number 3. The section where A and B are touching, the union of A and B.
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20:50:20 ** 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 --> Oh, I only shaded the middle section of both A and B. I would need to shade all of A and all of B.
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20:55:05 Query 2.3.100 Shade (A' ^ B) ^ C
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RESPONSE --> First I have to shade everything that is not A with B. That leaves me with B circle shaded, but not the part that is joined with A. Then I need to shade the area that comes into contact with C. The area that I shaded was the top right hand part of C, the area where only B and C are joined.
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20:55:35 ** 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 --> ok
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20:58:44 Describe the shading of the set (A ^ B)' U C.
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RESPONSE --> Shade everything except for A and B. Then shade C. Everything is shaded except for the upper part of A, the upper part of B and the middle touching part of A and B.
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21:00:21 ** 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 --> I forgot that it was a intersection of A and B, so I didn't shade the upper parts of A and B.
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21:05:42 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 don't know how to write these out. This is the section that I struggled with.
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21:09:27 ** 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 understand it when it is put as A - (B U C), but it is still difficult for me to try and think that way.
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