course Mth 151 V}ǛyP㴥vͯȀassignment #003
......!!!!!!!!...................................
19:16:03 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 --> {a, c, e, g} confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:16:13 **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 --> OK self critique assessment: 3
.................................................
......!!!!!!!!...................................
19:16:43 Give the intersection of the two sets Y and Z'
......!!!!!!!!...................................
RESPONSE --> {a} confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:16:51 **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'.**
......!!!!!!!!...................................
RESPONSE --> OK self critique assessment: 3
.................................................
......!!!!!!!!...................................
19:17:52 Query 2.3.30 describe in words (A ^ B' ) U (B ^ A')
......!!!!!!!!...................................
RESPONSE --> Find the intersection of A and B' (the elements they share). Find the intersection of B and A' (the elements they share). Find the union of these two sets. confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:18:17 ** 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 --> OK self critique assessment: 2
.................................................
......!!!!!!!!...................................
19:19:20 2.3.51 always or not always true: n(A U B) = n(A)+n(B)
......!!!!!!!!...................................
RESPONSE --> Not always true. confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:19:30 ** 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 --> OK self critique assessment: 3
.................................................
......!!!!!!!!...................................
19:22:34 Query 2.3.60 X = {1,3,5}, Y = {1,2,3}. Find (X ^ Y)' and X' U Y'.
......!!!!!!!!...................................
RESPONSE --> (X ^ Y') is {5} X' U Y' is {2, 4, 5} confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:23:49 ** 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 --> I misunderstood the first part of this question. I thought it was asking for (X ^ Y' ) instead of (X ^ Y)' self critique assessment: 2
.................................................
......!!!!!!!!...................................
19:27:41 2.3.72 A = {3,6,9,12}, B = {6,8}.
......!!!!!!!!...................................
RESPONSE --> n(A) = 4 n(B) = 2 So n(A * B) and n(B * A) are both equal to 4 * 2, or 8. confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:27:58 ** (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 --> OK self critique assessment: 3
.................................................
......!!!!!!!!...................................
19:29:36 2.3.84 Shade A U B
......!!!!!!!!...................................
RESPONSE --> I would shade in Circle A, Circle B, and the overlapping area between the two circles. confidence assessment: 2
.................................................
......!!!!!!!!...................................
19:29:44 ** 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 --> OK self critique assessment: 2
.................................................
......!!!!!!!!...................................
19:32:35 Query 2.3.100 Shade (A' ^ B) ^ C
......!!!!!!!!...................................
RESPONSE --> I would shade the area of Circle B that does not overlap with Circle A, and also shade the overlapping region between B and C (but not the area where A, B, and C all overlap). confidence assessment: 2
.................................................
......!!!!!!!!...................................
19:32:57 ** 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 --> OK self critique assessment: 2
.................................................
......!!!!!!!!...................................
19:34:43 Describe the shading of the set (A ^ B)' U C.
......!!!!!!!!...................................
RESPONSE --> Shade the part of Circle C that does not overlap with anything else. Also shade the area outside of the circles. confidence assessment: 2
.................................................
......!!!!!!!!...................................
19:35:12 ** 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 --> OK self critique assessment: 2
.................................................
......!!!!!!!!...................................
19:45:12 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 --> A - (B U C) confidence assessment: 2
.................................................
......!!!!!!!!...................................
19:45:26 ** 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 --> OK self critique assessment: 2
.................................................
"