course Mth 151
Q1: 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?
A1: In the first region you have what overlaps, or what the two sets have in common. In the region you would have a,c,f because they are the common numbers in the two. In the second region you have what is outside circle or what is left over from the first set that is not in region one, therefore you would have b, d, and e. In the third region you have what is left over from the overlapping region 1 in the second set of numbers which is g and k.
3
Ok
Q2: 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?
A2: In region 2 you would have 12 dark haired people without bright eyes because 8 people with bright eyes have dark hair. In region 3 you would have 7 people who have bright eyes and not dark hair because 8 people with bright eyes have dark hair. In region 1 you would have 8 people because 8 people have both dark hair and bright eyes. You then add 12+8+7= 27 people in the two circles which means that 8 people are not in the circle.
3
ok
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