course Mth 151
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20:23:23 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,b,c, ^ a,g U X a U a,c,e,g a,c,e,g,
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20:23:58 **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 --> ok
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20:25:51 Give the intersection of the two sets Y and Z'
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RESPONSE --> Y ^ Z' a,b,c ^ a,g a
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20:26:00 **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 -->
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20:26:05 **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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20:29:31 Query 2.3.30 describe in words (A ^ B' ) U (B ^ A')
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RESPONSE --> The set of elements that are in A and not in B or the set of elements that are in B and not in A.
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20:30:36 ** 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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20:32:16 2.3.51 always or not always true: n(A U B) = n(A)+n(B)
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RESPONSE --> Not always true because the sumof all th numbers (AUB) may not equal the sum of the numbers in A + the sum of the numbers in B.
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20:34:54 ** 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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20:38:22 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) (b) (X^Y)' X' U Y' 2,4,5 2,4 U 4,5 2, 4 5 For any element (X^Y)' = X' U Y'
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20:42:18 ** 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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20:45:42 2.3.72 A = {3,6,9,12}, B = {6,8}.
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RESPONSE --> AXB = { (3,6), (3,8), (6,6), (6,8), (9,6), (9,8), (12,6), (12,8)} BXA = { (6,3), (6,6), (6,9), (6,12), (8,3), (8,6), (8,9), (8,12) }
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20:46:51 ** (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 --> ok
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20:48:26 2.3.84 Shade A U B
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RESPONSE --> All of A and all of B should be shaded
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20:48:39 ** 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 --> ok
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20:51:59 Query 2.3.100 Shade (A' ^ B) ^ C
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RESPONSE --> The area that intersects between B and C should be shaded. Everything else should not.
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20:53:35 ** 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 --> This is hard for me to visualize without giving the Venn diagram numbers or letters. After I label the regions, I can better see what needs to be shaded. Will this be allowed on the test?
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20:57:01 Describe the shading of the set (A ^ B)' U C.
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RESPONSE --> Everything inside of A B and C should be shaded except for the area that is common to both A and B.
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20:57:48 ** 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 --> ok
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21:03:17 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' I hope this is right. I had to look in my text to find examples to arrive at this answer. It is much harder for me to work backwards and give the description of the shaded area.
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21:05:19 ** 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 am still having a difficult time visualizing this although I know understand which areas to shad if I am given the discription.
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