Math22QA

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course Mth 151

1/17/2012 7:41PM

002. Representing Sets

Note that there are 2 questions in this assignment.

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Question: `q001. We can represent the collection consisting of the letters a, b, c, d, e, f by a circle in which we write these letters. If we have another collection consisting of the letters a, c, f, g, k, we could represent it also by a circle containing these letters. If both collections are represented in the same diagram, then since the two collections have certain elements in common the two circles should overlap.

Sketch a diagram with two overlapping circles. The two circles will create four regions. The first region is the region where the circles overlap. The second region is the one outside of both circles. The third region is the part of the first circle that doesn't include the overlap. The fourth region is the part of the second circle that doesn't include the overlap. Number these regions with the Roman numerals I (the overlap), II (first circle outside overlap), III (second circle outside overlap) and IV (outside both circles).

Let the first circle contain the letters in the first collection and let the second circle contain the letters in the second collection, with the letters common to both circles represented in the overlapping region.

Which letters, if any, go in region I, which in region II, which in region III and which in region IV?

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Your solution:

A, C, and F would go in region I because they are in both collections. B, D, and E would go in region III because they are only in the first collection. G and K would go in region IV because they are only in the second group. Nothing would go in region II.

confidence rating #$&*: 3

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Given Solution:

The circles would appear and be labeled as below:

10The letters a, c and f go in the overlapping region, which we called Region I. The remaining letters in the first collection are b, d, and e, and they go in the part of the first circle that does not include the overlapping region, which we called Region II. The letters g and k go in the part of the second circle that does not include the overlapping region (Region III). There are no letters in Region IV.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q002. Suppose that 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.

Draw two circles, one representing the dark-haired people and the other representing the bright-eyed people. Represent the dark-haired people without bright eyes by writing this number in the part of the first circle that doesn't include the overlap (region II). Represent the number of bright-eyed people without dark hair by writing this number in the part of the second circle that doesn't include the overlap (region III). Write the appropriate number in the overlap (region I).

How many people are included in the first circle, and how many in the second?

How many people are included in both circles?

How many of the 35 people are not included in either circle?

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Your solution:

The first circle would have 12 because there are 20 dark hair people but 8 of them also have bright eyes so you take them out. The second circle would have 7 because there are 15 bright eye people but 8 of them have dark hair so you take them out. 8 people are included in both circles because they have both dark hair and bright eyes. 8 people would not be included in the circle because the first circle, second circle, and the overlap added together equals 27, and there are 35 people all together so if you subtract the total number of dark hair and bright eye people from the total people you get 8.

confidence rating #$&*: 3

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Given Solution:

These numbers are represented in the circles below.

• A number on the boundary of a circle indicates the total number in that circle, so the figure represents 20 individuals in circle A and 15 in circle B. The number 35 on the boundary of the entire figure represents the total number of individuals in the room, which in this case is 35.

• A number inside one of the regions I, II, III, IV represents the number in that region. The 8 having both dark hair and bright eyes will occupy the overlap between the circles (region I).

Of the 15 people with bright eyes, 8 also have dark hair so the other 7 do not have dark hair, and this number will be represented by the part of the second circle that doesn't include the overlap (region III). This is indicated by the number 7 inside region III in the figure below.

Of the 20 dark-haired people in the preceding example, 8 also have bright eyes. This leaves 12 dark-haired people for that part of the circle that doesn't include the overlap (region II).

We have accounted for 12 + 8 + 7 = 27 people. This leaves 35-27 = 8 people who are not included in either of the circles. The number 8 can be written outside the two circles (region IV) to indicate the 8 people who have neither dark hair nor bright eyes, as is indicated by the number 8 in region IV in the figure below:

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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