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21:56:28 `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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RESPONSE --> A ^ B are the people that have dark hair and bright eyes. Number of people in this collection are 8. A U B are the people who have dark hair or bright eyes. The number of people in this collection is 27.

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21:56:50 The collection A ^ B consists of all the people with both dark hair and bright eyes, which corresponds to the overlap between the two circles (region I). There are 8 people in this overlap, so we say n(A ^ B) = 8. The collection A U B consists of all the people who have least one of the characteristics. This would include the 12 people with dark hair but not bright eyes, located in the first circle but outside the overlap (region II); plus the 7 people with bright eyes but not dark hair, located in the second circle but outside the overlap (region III); plus the 8 people with both characteristics, located in the overlap (region I). Thus we include the 12 + 8 + 7 = 27 people who might be located anywhere within the two circles.

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RESPONSE --> ok

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22:01:16 `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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RESPONSE --> The characteristics of the people in A' are the people with bright eyes or neither dark hair or bright eyes. The characteristics of the people in B' are the people with dark hair or neither dark hair or bright eyes. n(A') = 15 n(B') = 20

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22:01:58 Of the 35 people, those in A' are those outside of A. Since A consists of all the dark-haired people, A' consists of all the people lacking dark hair. This includes the 8 people outside of both circles (people having neither dark hair nor bright eyes, region IV) and the 7 people in the second circle but outside the overlap (people having bright eyes but not dark hair, region III). n(A ' ) is therefore 8 + 7 = 15. Since B consists of all the bright-eyed people, B' consists of all the people lacking bright eyes. This would include the 8 people outside both circles (region IV), all of whom lack both dark hair and bright eyes, and the 12 people in the first circle but outside the overlap (region II), who have dark hair but not bright eyes. n ( B ' ) is therefore 12 + 8 = 20.

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RESPONSE --> ok

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22:07:43 `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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RESPONSE --> (A U B)' would be everyone outside the circles without either dark hair or bright eyes. n(A U B)' = 8 (A ^ B)' would be the people who have dark hair (first circle), the people who have bright eyes (second circle) and the people who have neither dark hair or bright eyes (outside both circles). This would not include the people who have both dark hair and bright eyes. n(A ^ B)' = 27

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22:09:08 A U B consists of everyone having at least one of the characteristics (dark hair, bright eyes), and is represented by the numbers in the two circles (regions I, II, III). ( A U B ) ' consists of the people who do not have at least one of the characteristics, and is represented by the number outside both circles (region IV). This number is 8, representing the 8 people who have neither dark hair nor bright eyes. A ^ B stands for all the people with both of the two characteristics (represented by the overlap, region I), so ( A ^ B ) ' stands for all the people who do not have both of the two characteristics (represented by everything outside region I, or regions II, III and IV). [ Note that (A ^ B)' is not the same as the collection of people who have neither characteristic. Anyone who does not have both characteristics will be in ( A ^ B ) ' . ] ( A ^ B )' must include those who have neither characteristic, and also those who have only one of the characteristics. The 8 people outside both circles, the 12 people in the first circle but outside the overlap, and the 7 people in the second circle but outside the overlap all lack at least one characteristic to, so these 8 + 12 + 7 = 27 people make up( A ^ B ) '.

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RESPONSE --> ok

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22:28:50 `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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RESPONSE --> A' U B' 7+8 U 12 + 8 15 U 20 = 35 A' are all the people with only bright eyes and the people that have neither dark hair or bright eyes. B' are all the people who have only dark hair or neither dark hair or bright eyes. The union of this would include the people with only bright eyes, neither dark hair or bright eyes and only dark hair. A' ^ B' 7+8 ^ 12+8 The intersection of this would be the people the people who have neither dark hair or bright eyes or 8 people.

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22:34:54 A ' U B ' consists of all the people who are in at least one of the sets A ' or B '. A ' consists of all the people who do not have dark hair, represented by every region of the diagram which does not include any of A. This will include the 7 people in B who are outside the overlapping region, and the 8 people who are outside of both A and B (regions III and IV. Since A consists of regions I and II, A' consists of regions III and IV). B ' consists of all the people who do not have bright eyes, represented by every region of the diagram which does not include any of B (regions II and IV). This will include the 12 people in A but outside the overlap, and the 8 people outside of both A and B. Thus A ' U B ' consists of everyone in at least one of A ' or B ', including the 7 people in B but outside the overlap (region III), the 12 people in A let outside the overlap (region II), and the 8 people outside of both A and B (region IV). These will be the people who lack at least one of the characteristics dark hair and/or bright eyes. Thus n(A' U B') = 7 + 12 + 8 = 27. Note that these are the same 27 people who are in ( A ^ B ) '. So at least in this case, ( A ^ B ) ' = A ' U B '. A ' ^ B ' consists of all the people in both A ' and B '. As before A ' includes the 7 people in B but not A (region III) as well as the 8 people outside both A and B (region IV), and B ' includes the 12 people in A but not B (region II) as well as the 8 people outside both A and B (region IV). The people in both A ' and B ' will be the 8 people outside both A and B, those who have neither dark hair nor bright eyes. We note that this is the same as the set ( A U B ) ', so at least for the present case we see that ( A ' ^ B ' = ( A U B ) '.

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RESPONSE --> You got me on the first one. I didn't think it through and added the eight people who had neither twice. I did figure it out by the second part though.

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22:48:20 `q005. Succinctly describe the relationships between ( A U B ) ', A ' U B ', (A ^ B) ' and A ' ^ B '.

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RESPONSE --> (A U B)' are all the the people in region IV that do who have neither dark hair or bright eyes A' U B' are all the people with only bright eyes (region III), only dark hair (region II) and neither dark hair or bright eyes (region IV). (A ^ B)' are all the people only dark hair (region II), only bright eyes (region III) and neither dark hair or bright eyes (region IV) So you could say that A' U B' = (A ^ B)' A' ^ B' are all the people with neither bright eyes or dark hair. So you could say that A' ^ B' = (A U B)'

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22:50:05 ( A U B ) ' = A ' ^ B ' and ( A ^ B ) ' = A ' U B '. The collection outside of the union A U B is the intersection A ' ^ B ', and the collection outside the intersection A ^ B is the union A ' U B '. The ' operation changes union to intersection and intersection to union.

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RESPONSE --> ok

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Good work. Let me know if you have questions.