course Mth 151 ??O1???U?????`?assignment #003003. `Query 3
......!!!!!!!!...................................
12:38:23 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?
......!!!!!!!!...................................
RESPONSE --> 2.3.15 is X' ^ Y' which is a,c confidence assessment: 3
.................................................
......!!!!!!!!...................................
12:38:44 **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. **
......!!!!!!!!...................................
RESPONSE --> Not the right question for 2.3.15 self critique assessment: 3
.................................................
......!!!!!!!!...................................
12:42:28 Give the intersection of the two sets Y and Z'
......!!!!!!!!...................................
RESPONSE --> Well Y is {a,b,c} and Z' is everything except {b,c,d,e,f} so the intersection is a confidence assessment: 2
.................................................
......!!!!!!!!...................................
12:42:50 **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'.**
......!!!!!!!!...................................
RESPONSE --> I forgot about g self critique assessment: 3
.................................................
......!!!!!!!!...................................
12:46:11 Query 2.3.30 describe in words (A ^ B' ) U (B ^ A')
......!!!!!!!!...................................
RESPONSE --> Well from the book it is 0 From here it is ( The group of A and the group of the complement of B have in common ?) is in union with ( The group of B and the group of the complement of A have in common ?) confidence assessment: 3
.................................................
......!!!!!!!!...................................
12:46:30 ** 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. **
......!!!!!!!!...................................
RESPONSE --> self critique assessment: 3
.................................................
......!!!!!!!!...................................
12:47:48 2.3.51 always or not always true: n(A U B) = n(A)+n(B)
......!!!!!!!!...................................
RESPONSE --> Out of the book 2.3.51 is always true However this particular problem on the computer is not always true confidence assessment: 3
.................................................
......!!!!!!!!...................................
12:47:58 ** 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. **
......!!!!!!!!...................................
RESPONSE --> self critique assessment: 3
.................................................
......!!!!!!!!...................................
12:52:27 Query 2.3.60 X = {1,3,5}, Y = {1,2,3}. Find (X ^ Y)' and X' U Y'.
......!!!!!!!!...................................
RESPONSE --> a. 2,4,5 b. 4 c. I really don't know what the conjecture would be confidence assessment: 1
.................................................
......!!!!!!!!...................................
12:53:11 ** 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'. **
......!!!!!!!!...................................
RESPONSE --> Ok I think I understand now. Unions confuse me to a certain degree. self critique assessment: 3
.................................................
......!!!!!!!!...................................
12:57:20 2.3.72 A = {3,6,9,12}, B = {6,8}.
......!!!!!!!!...................................
RESPONSE --> A x B= {(3x6)(3x8)(6x6)(6x8)(9x6)(9x8)(12x6)(12x8)} B x A= {(6x3)(6x6)(6x9)(6x12)(8x3)(8x6)(8x9)(8x12) confidence assessment: 3
.................................................
......!!!!!!!!...................................
12:57:36 ** (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) **
......!!!!!!!!...................................
RESPONSE --> self critique assessment: 3
.................................................
......!!!!!!!!...................................
12:58:23 2.3.84 Shade A U B
......!!!!!!!!...................................
RESPONSE --> The middle part of A and B would be shaded confidence assessment: 3
.................................................
......!!!!!!!!...................................
12:58:43 ** 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. **
......!!!!!!!!...................................
RESPONSE --> I thought it would just be the middles? self critique assessment: 3
.................................................
......!!!!!!!!...................................
12:59:50 Query 2.3.100 Shade (A' ^ B) ^ C
......!!!!!!!!...................................
RESPONSE --> Whatever is not in A but is contained in both B and C should be shaded confidence assessment: 2
.................................................
......!!!!!!!!...................................
13:00:05 ** 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 **
......!!!!!!!!...................................
RESPONSE --> I got it! self critique assessment: 3
.................................................
......!!!!!!!!...................................
13:00:40 Describe the shading of the set (A ^ B)' U C.
......!!!!!!!!...................................
RESPONSE --> It would be whatever is outside both A and B and includes C confidence assessment: 3
.................................................
......!!!!!!!!...................................
13:00:51 ** 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. **
......!!!!!!!!...................................
RESPONSE --> self critique assessment: 3
.................................................
......!!!!!!!!...................................
13:01:35 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.
......!!!!!!!!...................................
RESPONSE --> ( B ^C)' U A confidence assessment: 2
.................................................
......!!!!!!!!...................................
13:01:54 ** 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. **
......!!!!!!!!...................................
RESPONSE --> self critique assessment:
.................................................
......!!!!!!!!...................................
13:01:57 ** 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. **
......!!!!!!!!...................................
RESPONSE --> self critique assessment: 3
.................................................