Assignment 02

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course MTH 151

12:42pm, 1/28/14

If your solution to stated problem does not match the given solution, you should self-critique per instructions at

http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm

Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

002. Representing Sets

Note that there are 3 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:

In Region I, letters a, c and f overlap, which is why they are in region I. In Region II, letters b, d and e are not overlapping, and would go in this region for this reason. The letters g and k are not in the first collection, so they would go in Region III. No letters are in Region IV, because there are no letters left.

confidence rating #$&*: 3

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Given Solution:

The circles would appear and be labeled as below:

The 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):

Had to draw this out once, and then a second time, to make sure that I didn’t leave any letters out.

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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 20 people, because there are 20 people that only have dark hair. This area would be Region II. In the second circle, there are 15 people in the second circle, because there are 15 people with bright eyes only. There are 8 people included in both circles, because there are 8 people with both dark hair and bright eyes. All people are included in either circle.

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):

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Self-critique Rating: 3

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Question: `q003. Suppose there are 200 people in a hall, 140 having dark hair, 90 having short hair and 50 having hair which is neither dark nor short. Sketch a diagram like the ones above, specify how many people are in each of the four regions and describe the people in each region.

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Your solution:

In the first circle, and Region I, we would place the 200 people. In the second circle, and Region II, we would place the 140 dark haired people. In Region III, we could put the people that have dark hair and short hair. In Region IV, we would put the 50 people that don’t have dark hair or short hair. Region III would be the overlapping area of the 2 circles.

confidence rating #$&*:

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Self-critique Rating: 3

Self-critique (if necessary):

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Self-critique rating:

Self-critique (if necessary):

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Self-critique rating:

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