course Mth 151 I seem to still be confused as to what exactly is being asked and what things are included in each group. »¯°´µçòs«“ãzÕ€ÞÀäïH¸Ë]¬ÞÕassignment #003
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09:05:09 `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 --> The collection A ^ B includes the 8 people who have dark hair and bright eyes, due to these people having both characteristics common for both sets. The collection A U B contains 35 people, due to a union being a set of all elements that belongs to either set. confidence assessment: 1
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09:06:16 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 --> I realize my mistake on this second problem. I have added the sets by 20 and 15 respectively, and did not subtract the number of people from each set with only 1 of the elements. self critique assessment: 2
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09:08:54 `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 people that are in A' indicate that they have bright eyes and do not have dark hair. The characteristics of people in B' indicate people with dark hair, yet do not have bright eyes. The n(A') = 7, and n(B') = 12. confidence assessment: 1
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09:10:36 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 --> Again I realize my confusion because the people in set A' include all the people with bright eyes, regardless of the fact that they may also have dark hair. Same for B' which includes all people with dark hair, regardless of the fact they may also have bright eyes. self critique assessment: 2
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09:13:47 `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 --> The characteristics for people outside A U B means that they do not have dark hair or bright eyes, and in this collection, they are no people who meet this qualification. The people in A ^ B include 8 people who have either dark hair and bright eyes. confidence assessment: 1
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09:22:18 `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 --> The set of A' U B' includes the people not in A and not in B. The number of people not in A is 7, and the number of people not in B is 12. The number of people not in A or B is 8, due to them both having dark hair and light eyes. confidence assessment: 0
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09:26:16 `q005. Succinctly describe the relationships between ( A U B ) ', A ' U B ', (A ^ B) ' and A ' ^ B '.
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RESPONSE --> The relationships between (A U B )' includes all of the people who have neither characteristic, A' U B' includes the people who have neither characteristic of A or B, and (A ^ B)' includes people who have have neither characteristic. A' ^ B' includes people who have neither characteristic of A or B. confidence assessment: 0
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09:26:59 ( 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 --> Perhaps the wording of these problems confuses me. I have done the hw but apparently I still lack understanding on what is being asked. self critique assessment: 1
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ñáü…žÚä£Õ黋°‚íÄ£G§zÄœÄJzÎzŒ» assignment #003 003. Intersection, Union, Complement, de Morgans Laws Liberal Arts Mathematics I 02-01-2009
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