course MTH 151
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23:40:12 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 --> # 19 from the book: Z' = {a,g} so (Y ^ Z') = {a,b,c} ^ {a,g} = {a} {a} U X = {a,c,e,g}
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23:40:30 **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 --> right
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23:41:30 Give the intersection of the two sets Y and Z'
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RESPONSE --> Y ^ Z' = {a}
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23:41:40 **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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23:42:48 Query 2.3.30 describe in words (A ^ B' ) U (B ^ A')
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RESPONSE --> The set of all elements in A and not in B, or are in B but not in A.
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23:43:27 ** 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 --> right
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23:43:51 2.3.51 always or not always true: n(A U B) = n(A)+n(B)
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RESPONSE --> not always
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23:45:22 ** 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 --> ok
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23:48:48 Query 2.3.60 X = {1,3,5}, Y = {1,2,3}. Find (X ^ Y)' and X' U Y'.
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RESPONSE --> A rule in our book states that (A ^ B)' = A' U B', hence (X ^ Y)' = X' U Y' X' U Y' = {2,4,5}
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23:49:15 ** 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 --> ok
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23:51:31 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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23:52:34 ** (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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23:53:12 2.3.84 Shade A U B
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RESPONSE --> on my diagram all of A is shaded as is all of B
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23:53:21 ** 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 --> right
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23:56:04 Query 2.3.100 Shade (A' ^ B) ^ C
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RESPONSE --> The only areas shaded are the overlap between B and C and the area of C not included in A or the overlap b/w A and B. I can't draw on here so I hope my exlpanation makes sense.
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23:56:42 ** 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 --> got it!
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23:59:42 Describe the shading of the set (A ^ B)' U C.
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RESPONSE --> (A ^ B)' = A' U B' I think the only area shaded is the part of C that doesn't overlap with any other region.
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00:02:44 ** 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 made this more complicated than neccessary. I drew the correct diagram and will go over it.
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00:03:09 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 --> A ^ B' ^ C'
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00:04:14 ** 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 --> ok, so mine is correct?
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