2.4 Venn Diagrams with Three Sets and Verification of Equality of Sets 73 Venn diagrams can be used to illustrate and analyze many everyday problems. One example follows. Example 2 Blood Types Human blood is classified (typed) according to the presence or absence of the specific antigens A, B, and Rh in the red blood cells. Antigens are highly specified proteins and carbohydrates that will trigger the production of antibodies in the blood to fight infection. Blood containing the Rh antigen is labeled positive, ,+ while blood lacking the Rh antigen is labeled negative, .− Blood lacking both A and B antigens is called type O. Sketch a Venn diagram with three sets A, B, and Rh and place each type of blood listed in the proper region. A person has only one type of blood. Solution As illustrated in Chapter 1, the first thing to do is to read the question carefully and make sure you understand what is given and what you are asked to determine. There are three antigens A, B, and Rh. Therefore, begin by naming the three circles in a Venn diagram with the three antigens; see Fig. 2.18. Any blood containing the Rh antigen is positive, and any blood not containing the Rh antigen is negative. Therefore, all blood in the Rh circle is positive, and all blood outside the Rh circle is negative. The intersection of all three sets, region V, is AB+. Region II contains only antigens A and B and is therefore − AB . Region I is A− because it contains only antigen A. Region III is B ,− region IV is A+, and region VI is B+. Region VII is O ,+ containing only the Rh antigen. Region VIII, which lacks all three antigens, is −O . 7 Now try Exercise 71 Positive blood O1 (no A or B antigens) AB1 B1 A1 A antigen Rh antigen Red blood cell Negative blood (no Rh antigen) O2 AB2 B2 A2 Red blood cell B antigen I III IV VI VII VIII U A B II V A2 B2 AB2 AB1 A1 B1 Rh O1 O2 Figure 2.18 Now determine what elements go in region IV. >A C {5, 9} = Since 5 and 9 have already been placed in region V, there are no elements to be placed in region IV. Now determine the elements to go in region VI. >B C {1, 5, 8, 9} = Since both the 5 and 9 have already been placed in region V, place the 1 and 8 in region VI. Now complete set A. The only element of set A that has not previously been placed in regions II, IV, or V is 12. Therefore, place the element 12 in region I. The element 12 that is placed in region I is only in set A and not in set B or set C. Using set B, complete region III using the same general procedure used to determine the elements in region I. Using set C, complete region VII by using the same procedure used to complete regions I and III. To determine the elements in region VIII, determine the elements in U that have not been placed in regions I–VII. The elements 6 and 7 have not been placed in regions I–VII, so place them in region VIII. 7 Now try Exercise 5
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