# { x / x A or x B} A B. That s important, the set union is defined by shading in both circles of the Venn Diagram, all the members belong.

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1 VENN DIGRMS In the last section we represented sets using set notation, listing the elements in brackets. With Venn Diagrams, we define everything exactly the same way, but the definitions are in the form of pictures. For instance, the set union of and, written was defined as: = { x / x or x } With Venn Diagrams, the definition again is all the members that belong to or, but we show that by shading in the circles and. That s important, the set union is defined by shading in both circles of the Venn Diagram, all the members belong. Let s look at the set intersection of and, that was defined as the members that belonged to both sets. = { x / x and x } That is illustrated by shading only the portion of the circles that overlap. Those elements belong to and also to.

2 Notice, there were rectangles around those examples. Those rectangles represented the universal set. ll the elements under discussion. To represent a single set, such as. I would draw one circle and shade it in. How would the complement of, - be illustrated? I m glad you asked. Just like before, the complement of are all the members of the universal set not in. = { x / x and x } That s illustrated by shading in the rectangle, but not any part of the circle. I know, you love coloring, you want to do more. Let s look at the set difference. Remember was defined as all the elements in, but couldn t belong to. = { x/ and x } Venn Diagrams -2

3 Having those operations defined through Venn Diagrams, we can now play with more sets. gain, the illustrations are important, introducing a third set using a circle does not in any way change what we have already defined. In fact, if there is a third set, we work with just two at a time, just like we did with sets. Example Shade C First we ll find the intersection of and, completely ignoring C. fter that, we take that result and intersect it with C. We already know be definition, that looks like this Now, we ll take that shading and intersect that with circle C. I shaded with horizontal lines and C with vertical so you can tell the difference. Where the shading overlaps is what these sets have in common. That almost makes sense, for an element to belong to, and C, there is only one region within the three circles that satisfies that. C Venn Diagrams -3

4 Let s look at a Venn Diagram made up of three sets in which the regions are labeled. Now, we ll describe each region C Region 5 is in all three circles. So any elements in region 5 would belong to all three sets. In other words; C. What about Region 2? Those are the elements in and, but not C. How might you describe Region 6? Those are elements in and C, but not in. Try Region 4. The elements in and C, but not. This is fun, let s look at some more regions. Region 1 describes the elements in only. What about region 3? Those elements are only in. Region 7 then would be the elements in C only. Region 8 would describe elements that are not members of any of the sets, but belong to the universal set. It s important that you become familiar with how each of those regions might be described. eing able to describe those regions would allow you to solve some problems. Example survey was taken of 650 university students. It was reported that 240 were taking math, 290 were taking biology, and 270 were enrolled in chemistry. Of those students, 80 were taking biology and math, 70 were taking math and chemistry, 60 were taking biology and chemistry, and 50 were taking all three classes. How many students took math only? t first glance, you might not think this is possible because the numbers add up to more than 650. ut if you are familiar with how the regions are described, we can determine how many were in each region. Venn Diagrams -4

5 In going about this problem, I would tell you to draw a Venn Diagram and begin by filling in Region 5, the students that took all three courses. M 50 C fter doing that, we ll place the number of students taking each course on the circle because we don t know where those students should be located within the circles M 50 C 270 Okay, now we can have some fun by determining what regions the students should be located. For instance, it says that 80 students are taking math and biology. We have 50 of those accounted for in Region 5, how many does that leave to be in Region 2? = 30 That s easy enough. sing that same reasoning, 70 students are taking math and chemistry, how many students would then be in Region 4? Well, = 20. That s pretty easy, don t you think? Ok, how many students should be in Region 6? Since there are 60 students enrolled in biology and chemistry, and 50 of them are accounted for in Region 5, that leaves 10 students for Region 6. Venn Diagrams -5

6 Let s fill in those numbers: M C 270 Now how many students would be in Region 1? Now remember, there are supposed to be 240 students taking math, we have 100 accounted for in Regions 2, 4, and 5. That leaves 140 students in region 1, taking math only. How many students are taking biology only? Well, we were told that 290 students were taking biology, we have 90 of them accounted for in Regions 2, 5, and 6, that leaves 200 students in Region 3. How many are taking only chemistry? We know there are 270 students taking chemistry, we have 90 accounted for in Regions 4, 5, and 6, that leaves 190 in Region 7. Filling in those numbers and taking the numbers off the circle, we have the following information. M C We have one slight problem, if we add those regions within the circles, the total is 640 students. The problem stated 650 were surveyed, we re missing ten students. Where are they? That s right, they would be in Region 8, not taking any of those courses. Now tell me, was that fun? How many students took math and biology, but not chemistry? How many students took math and chemistry, but not biology? How many students took biology and chemistry, but not math? How many students took only math? How many students took exactly two of the courses? Venn Diagrams -6

10 VENN DIGRMS 13a. t a pow-wow in rizona, Native mericans from all over the Southwest came to participate in the ceremonies. coordinator of the pow-wow took a survey and found that: 15 families brought food, costumes, and crafts: 25 families brought food and crafts: 42 families brought food: 20 families brought costumes and food: 6 families brought costumes and crafts, but not food: 4 families brought crafts, but neither food nor costumes: 10 families brought none of the three items: 18 families brought costumes, but not crafts: (b) (c) (d) (e) How many families were surveyed? How many families brought costumes? How many families brought crafts, but not costumes? How many families did not bring crafts? How many families brought food or costumes? 14a. survey of people attending a Lunar New Year celebration in Chinatown yielded the following results: 120 were women: 150 spoke Cantonese: 170 lit firecrackers: 108 of the men spoke Cantonese: 100 of the men did not light firecrackers: 18 of the non-cantonese-speaking women lit firecrackers: 78 non-cantonese-speaking men did not light firecrackers: 30 of the women who spoke Cantonese lit firecrackers. (b) (c) (d) How many attended? How many of those who attended did not speak Cantonese? How many women did not light firecrackers? How many of those who lit firecrackers were Cantonese-speaking men? Venn Diagrams -10

12 VENN DIGRMS 17. survey of 80 sophomores at a western college showed that: 36 took English: 32 took history: 32 took political science: 16 took political science and history: 16 took history and English: 14 took political science and English: 6 took all three How many students: (b) (c) (d) (e) took English and neither of the other two? took none of the three courses? took history, but neither of the other two? took political science and history, but not English? did not take political science? Venn Diagrams -12

13 Venn Diagrams -13

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