Query 3

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course Mth 151

003.  `Query 3 

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Question:  `qQuery  2.3.15  This might differ from the problem as given in the text, but you should be able to answer it for the given sets: universal set U = {a,b, c,…,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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Your solution: 

Z’ = {A, G}, so (Y ^ Z’) = {A}

So (Y ^ Z’) U X = {A, C, E, G}

 

 

confidence rating #$&*: 3

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Given Solution: 

`a**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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Self-critique (if necessary):Ok

 

 

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Self-critique Rating: 3

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Question:  `qGive the intersection of the two sets Y and Z'

 

 

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Your solution: 

 Y ^ Z’ = {A}

 

 

confidence rating #$&*: 3

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Given Solution: 

`a**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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Self-critique (if necessary): ok

 

 

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Self-critique Rating: 3

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Question:  `qQuery  2.3.32 (formerly 2.3.30).  This was not assigned, but you answered a series of similar questions and should be able to give a reasonable answer to this one:  Describe in words (A ^ B' ) U (B ^ A')

 

 

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Your solution:  All elements in A that are not in B, or all elements in B that are not in A

 

 

confidence rating #$&*: 2

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Given Solution: 

`a**   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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Self-critique (if necessary): Ok

 

 

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Self-critique Rating: 3

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Question:  `q2.3.53 (formerly 2.3.51) Is it always or not always true that n(A U B) = n(A)+n(B)?  This was not among the assigned questions but having completed the assignment you should be able to answer this.

 

 

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Your solution:  False. There are several cases in which the two sets that go into the first union would not emerge with the same results in the second pairing.

 

 

confidence rating #$&*: 3

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Given Solution: 

`a** 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.

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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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Self-critique (if necessary): Ok

 

 

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Self-critique Rating: 3

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Question:  `qQuery 2.3.60   X = {1,3,5}, Y = {1,2,3}.  Find (X ^ Y)' and X' U Y'.

 

 

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Your solution: 

( X ^ Y )' = {2, 4, 5}

X' U Y' = {2, 4, 5}

 

 

confidence rating #$&*: 2

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Given Solution: 

`a** 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'. **

 

STUDENT QUESTION:

 

Where did the 4 come from?

INSTRUCTOR RESPONSE:

 

I believe this problem, as stated in the text, indicates that the universal set is {1, 2, 3, 4, 5}.

 

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Self-critique (if necessary): ok

 

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Self-critique Rating: 3

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Question:  `q2.3.72  A = {3,6,9,12}, B = {6,8}.

 

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Your solution: 

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)

 

 

confidence rating #$&*: 3

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Given Solution: 

`a**  (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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Self-critique (if necessary): ok

 

 

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Self-critique Rating: 3

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Question:  `q2.3.84  Shade A U B

 

 

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Your solution:  Shaded entirety of circle A and circle B, but not the rectangle they're contained in.

 

 

confidence rating #$&*: 3

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Given Solution: 

`a** 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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Self-critique (if necessary): ok

 

 

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Self-critique Rating: 3

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Question:  `qQuery 2.3.100 Shade (A' ^ B) ^ C

 

 

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Your solution:  Shaded circles B and C, except for where they overlap with A, as well as region outside circles.

 

 

confidence rating #$&*: 3

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Given Solution: 

`a** 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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Self-critique (if necessary): ok

 

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Self-critique Rating: 3

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Question:  `qQuery 2.3.108.  Describe the shading of the set (A ^ B)' U C.

 

 

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Your solution:  Shaded all regions except the intersection between A and B

 

 

confidence rating #$&*: 3

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Given Solution: 

`a** 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.  **

 

 

STUDENT QUESTION

 

 I think I understand because the ‘ was outside the ( ) then only the answer to A^B would be prime. And so my answer is 
wrong to the extent that the larger regions of A &B would also be shaded, but had it been (AUB)’ no part of either A or B 
would have been Shaded?

INSTRUCTOR RESPONSE

 

Exactly. Very good question, which you answered very well.

 

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Self-critique (if necessary): ok

 

 

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Self-critique Rating: 3

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Question:  `q2.3.114 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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Your solution:  {C' ^ B'} ^A

 

 

confidence rating #$&*: 3

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Given Solution: 

`a** 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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Self-critique (if necessary):

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Self-critique rating:

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Question:  `q2.3.114 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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Your solution:  {C' ^ B'} ^A

 

 

confidence rating #$&*: 3

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Given Solution: 

`a** 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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Self-critique (if necessary):

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Self-critique rating:

#*&!

&#Very good responses. Let me know if you have questions. &#