course Mth 151

6/6 7

Question: `q001. Note that there are 5 questions in this assignment.

Again we have a total of 35 people in a room. Of these, 20 have dark hair and 15 have bright eyes. There are 8 people with dark hair and bright eyes.

Let A stand for the collection of people who have dark hair and B for the collection who have bright eyes. The Intersection of these two collections is denoted A ^ B, and stands for the collection of all people who have both dark hair and bright eyes. The Union of these two collections is denoted A U B, and stands for the collection of all people who have at least one of these characteristics.

In terms of the diagram you made for the preceding problem, describe the collection A ^ B and the collection A U B. Give the number of people in each of these collections (these numbers are designated by the notation n ( A ^ B) and n(A U B) ). Refer to the diagrams you have made.

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Your solution:

A ^ B is the collection of all people who have both dark hair and bright eyes and are shown in the overlapping region of the diagram, n(A ^ B) = 8. A U B is the collection of all people who have at least one of these characteristics which includes the 12 people in the first circle who only have dark hair, the 7 people in the second circle who only have bright eyes, as well as the 8 people in the overlapping region who have both characteristics, 12 + 7 + 8 = 27.

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Question: `q002. Continuing the preceding example, we let A' stand for the people who are not in the collection A, and we let B' stand for the people who are not in the collection B.

What are the characteristics of the people in A', and what characterizes people in B' ? What are n(A ') and n(B '), the numbers of people in A' and B' ?

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Your solution:

Since A ‘ is the people who are not in collection A, n(A ‘)= 15 people who have bright eyes only. These are the 7 people located inside of the second circle and the 8 people who are located outside of the circle who do not possess either characteristic. B ‘ is the people who aren’t in collection B which would be the 12 people in the first circle who have only dark hair and the 8 people who are outside of both circles who possess neither characteristic, n(B ‘) = 20

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Question: `q003. ( A U B ) ' stands for the everyone outside A U B, and ( A ^ B ) ' stands for everyone outside A ^ B. What characterizes the people in each of these collections, and how many people are there in each?

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Your solution:

(A U B) ‘ includes the people who do not have at least one of the characteristics which is represented by the number outside of both circles, 8. (A B) ‘ consists of anyone who doesn’t have both characteristics which is represented by the 12 people in the first circle who have only dark hair, the 7 people in the second circle who have only bright eyes, and the 8 people outside of both circles who have neither of the characteristics, 12 + 7 + 8 + 27.

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Question: `q004. How many people are in A ' U B ', and how could those people be characterized? Answer the same for A ' ^ B '.

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Your solution:

A ‘ UB ‘ contains anyone who has at least one of the characteristics of A ‘ or B ‘, n(A’ U B’) = 27. A ‘ is the people who are not in collection A who have bright eyes only. These are the 7 people located inside of the second circle and the 8 people who are located outside of the circle who do not possess either characteristic. B ‘ is the people who aren’t in collection B which would be the 12 people in the first circle who have only dark hair and the 8 people who are outside of both circles who possess neither characteristic. n(A’ ^ B’) = 27

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Question: `q005. Succinctly describe the relationships between ( A U B ) ', A ' U B ', (A ^ B) ' and A ' ^ B '.

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Your solution:

(A U B)’ is the same as A’^ B’ and A ‘ U B’ is the same as (A ^ B )’.

&#I need to see the questions so I can be sure what your answers mean. Most of the time I can tell, but I'm dealing with information that comes in from over 1000 different files, containing a total of about 10 000 questions. While I'm familiar with the content and sequencing of the questions, having written them all, and know what I'm looking for, different students will answer these questions in different ways and I need to be able to relate your answers to the specific wording of each question.

When reviewing my responses you will also need to be able to relate your answers and my comments to the specifics of the original document.

So it will be important for you on future documents to insert your responses into a copy of the original document, according to instructions, without otherwise changing any of the content of the original document.

This will ensure you of the best possible feedback on your work. &#

#$&*

Due to a carelessly typed password, a block of files failed to post correctly on 6/7/10. This file might have been among them, and is being posted on 6/8/10, a day late.

course

What is the set (Y ^ Z')U X?

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Your solution:

Z’= {a,g}, Y^Z={a}

(Y ^ Z’)U X= {a, c, e, g}

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

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Your solution:

Z’= {a, g}

The intersection of sets Y and Z’ is {a} because it is the only common element in both sets.

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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:

The set of all elements that are in A but not in B, or that are in B but not in A.

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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:

This is not always true because it depends on the sets A and B.

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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= {1,3} which means (X^Y)’ = {1,3}’ = {2, 4, 5}

X’ = {2, 4}, Y’ = {4, 5}, so (X’ U Y’) = {2, 4, 5}

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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)}

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Question: `q2.3.84 Shade A U B

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Your solution:

Everything in A and B would be shaded whereas the region outside of the circles but still in the box wouldn’t be shaded.

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

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Your solution:

The shaded region includes the inside of B and C as well as the overlap of B and C and what lies outside of A. No part of region A should be shaded, including overlapping regions.

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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:

The only region not shaded is the overlap of A and B that isn’t a part of C. Every part of C, the non-overlapping parts of A and B and everything outside of A and B is shaded.

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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:

A – ( B U C )

&#I need to see the questions so I can be sure what your answers mean. Most of the time I can tell, but I'm dealing with information that comes in from over 1000 different files, containing a total of about 10 000 questions. While I'm familiar with the content and sequencing of the questions, having written them all, and know what I'm looking for, different students will answer these questions in different ways and I need to be able to relate your answers to the specific wording of each question.

When reviewing my responses you will also need to be able to relate your answers and my comments to the specifics of the original document.

So it will be important for you on future documents to insert your responses into a copy of the original document, according to instructions, without otherwise changing any of the content of the original document.

This will ensure you of the best possible feedback on your work. &#

#$&*

Due to a carelessly typed password, a block of files failed to post correctly on 6/7/10. This file might have been among them, and is being posted on 6/8/10, a day late.