Assignment 03

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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:

(Y ^ Z') U X = {a, c, e, g}

Z' = {a,g} 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'.

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):

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Self-critique Rating: 2

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

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Your solution:

intersection of Y and Z' is {a} because it is the only common element to both sets.

confidence rating #$&*: 2

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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): I am a little confused about finding the intersections.

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The intersection of two sets is the set of all elements that are present in both sets.

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Self-critique Rating: 2

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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:

(A^B') U (B^A') means all of the elements that are in A and not B, or all of the elements that are in B and not A.

confidence rating #$&*: 3

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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):

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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:

It is not always true. n(A U B) does not always equal n(A)+n(B) if there are some of the same elements in both sets.

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):

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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} because X ^ Y = {1,3}, so {1,3}' = {2,4,5}

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

confidence rating #$&*: 3

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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):

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Self-critique Rating: 3

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Question: `q2.3.72 A = {3,6,9,12}, B = {6,8}. What is A X B and what is n(A X B)?

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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)}

n(A X B) = {(6,3), (6,6), (6,9), (6,12), (8,3), (8,6), (8,9) (8,12)}

confidence rating #$&*: 2

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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):

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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:

All of A and B is shaded. Everything outside of the rectangle is not.

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):

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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:

I shaded every region that included B and C, but no part of A.

confidence rating #$&*: 2

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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):

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Self-critique Rating: 2

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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:

All of C is shaded. All of A and B would be shaded, except the area where they overlap. All portions outside of A and B would be shaded as well.

confidence rating #$&*: 2

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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): 2

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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:

(B' ^ c') ^A

confidence rating #$&*: 2

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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:

(B' ^ c') ^A

confidence rating #$&*: 2

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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:

#*&!

&#Good responses. See my notes and let me know if you have questions. &#