MTH-151 chapter 2

course MTH 151

i hope i did this right. i printed off the test for chapter 2. i apologize i am falling so behind.

>**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 --> >good >................................................. >......!!!!!!!!................................... > >14:35:02 >Give the intersection of the two sets Y and Z' >......!!!!!!!!................................... >RESPONSE --> >{a} >................................................. >......!!!!!!!!................................... > >14:35:12 >**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 --> >good >................................................. >......!!!!!!!!................................... > >14:36:35 >Query 2.3.30 describe in words (A ^ B' ) U (B ^ A') >......!!!!!!!!................................... >RESPONSE --> >this means all elements that are in both A and opposite B >unified with all elements that are in both band and opposite >A >................................................. >......!!!!!!!!................................... > >14:37:01 >** 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 >................................................. >......!!!!!!!!................................... > >14:37:53 >2.3.51 always or not always true: n(A U B) = n(A)+n(B) >......!!!!!!!!................................... >RESPONSE --> >always true >................................................. >......!!!!!!!!................................... > >14:38:07 >** 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 >................................................. >......!!!!!!!!................................... > >14:41:24 >Query 2.3.60 X = {1,3,5}, Y = {1,2,3}. Find (X ^ Y)' and >X' U Y'. >......!!!!!!!!................................... >RESPONSE --> >(a) {4} (b) {2,4,5} >................................................. >......!!!!!!!!................................... > >14:41:43 >** 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 >................................................. >......!!!!!!!!................................... > >14:46:08 >2.3.72 A = {3,6,9,12}, B = {6,8}. >......!!!!!!!!................................... >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). >................................................. >......!!!!!!!!................................... > >14:46:16 >** (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 --> >good >................................................. >......!!!!!!!!................................... > >14:46:44 >2.3.84 Shade A U B >......!!!!!!!!................................... >RESPONSE --> >i do not know how to draw with this thing >................................................. >......!!!!!!!!................................... > >14:46:57 >** 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 --> >yes >................................................. >......!!!!!!!!................................... > >14:49:01 >Query 2.3.100 Shade (A' ^ B) ^ C >......!!!!!!!!................................... >RESPONSE --> >C intersected with B would be shaded but not any of A would >be shaded >................................................. >......!!!!!!!!................................... > >14:49:08 >** 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 >................................................. >......!!!!!!!!................................... > >14:49:51 >Describe the shading of the set (A ^ B)' U C. >......!!!!!!!!................................... >RESPONSE --> >do not shade any of A or B but all of C that was not >intersected with a or b >................................................. >......!!!!!!!!................................... > >14:50:05 >** 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 >................................................. >......!!!!!!!!................................... > >14:50:35 >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 --> >i do not understand this >................................................. >......!!!!!!!!................................... > >14:51:34 >** 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 so you are wanting us to explain to you how that would be >if it were a question >................................................. > cD騆xY >assignment #004 >\ҒMҶ̲Ϯ_ >Liberal Arts Mathematics I >02-13-2006 >......!!!!!!!!................................... > >15:12:34 >2.4.12 n(A') = 25, n(B) = 28, n(A' U B') = 40, n(A ^ B) = 10 >......!!!!!!!!................................... >RESPONSE --> > i don't know >................................................. >......!!!!!!!!................................... > >15:13:55 >** In terms of the picture (2 circles, linked, representing >the two sets) there are 28 in B and 10 in A ^ B so there are >18 in the region of B outside of A--this is the region B-A. > >There are 25 outside of A, and 18 of these are accounted for >in this region of B. Everything else outside of A must >therefore also be outside of B, so there are 25-18=7 >elements in the region outside of both A and B. > >A ' U B ' consists of everything that is either outside of A >or outside of B, or both. The only region that's not part >of A ' U B ' is therefore the intersection A ^ B, since >everything in this region is inside both sets. > >A' U B' is therefore everything but the region A ^ B which >is common to both A and B. This includes the 18 elements in >B that aren't in A and the 7 outside both A and B. This >leaves 40 - 18 - 7 = 15 in the region of A that doesn't >include any of B. This region is the region A - B you are >looking for. ** >......!!!!!!!!................................... >RESPONSE --> >ok so where do we use the 40 the book refers ti

I believe that n(A' U B') = 40.

>................................................. >......!!!!!!!!................................... > >15:15:45 >query 2.4.18 wrote and produced 2, wrote 5, produced 7 >&&&& How many did he write but not produce? >......!!!!!!!!................................... >RESPONSE --> >3 >................................................. >......!!!!!!!!................................... > >15:15:53 >** You need to count the two he wrote and produced among >those he wrote, and also among those he produced. > >He only wrote 5, two of which he also produced. So he wrote >only 3 without producing them. > >In terms of the circles you might have a set A with 5 >elements (representing what he wrote), B with 7 elements >(representing what he produced) and A ^ B with 2 elements. >This leaves 3 elements in the single region A - B and 5 >elements in the single region B - A. The 3 elements in B - >A would be the answer to the question. ** >......!!!!!!!!................................... >RESPONSE --> >good >................................................. >......!!!!!!!!................................... > >15:20:52 >2.4.24 9 fat red r, 18 thn brown r, 2 fat red h, 6 thin red >r, 26 fat r, 5 thin red h, 37 fat, 7 thin >brown ......!!!!!!!!................................... >......!!!!!!!!................................... >RESPONSE --> >ok >................................................. >......!!!!!!!!................................... > >15:21:42 >** Here's my solution. Tell me if there is anything you >disagree with (I'm not infallible) or don't understand. > >incidental: 18 thin brown roosters, 7 thin brown hens, 6 >thin red hens and the 6 thin roosters which aren't fat (out >of the 50-26=24 thin roosters 18 are brown so 6 are red) >adds up to 37 thin chickens > >How many chickens are fat? > >37 as given > >How many chickens are red? > >22: 9 fat red roosters, 6 thin red roosters, 5 thin red >hens, 2 fat red hens. > > > >How many chickens are male? > >50: 9 fat red roosters are counted among the 26 fat roosters >so the remaining 17 fat roosters are brown; then there are >18 thin brown roosters and 6 thin red roosters; the number >of roosters therefore adds up to 9 + 18 + 6 + 17 = 50 > >How many chickens are fat not male? > >26 of the 37 fat chickens are male, leaving 11 female > >How many chickens are brown not fat? > >25: 18 thin brown roosters, 7 thin brown hens adds up to 25 >thin brown chickens > >How many chickens are red and fat? > >11: 9 fat red roosters and 2 fat red hens.** >......!!!!!!!!................................... >RESPONSE --> >ok i have looked at this prob numerous times and each time >come up with a different answer >................................................. > "

You appear to understand most of these ideas. However, as noted before, you need to explain many of your solutions more fully. You aren't providing enough detail for me to tell for sure what you do and do not understand.

Let me know if you have questions.