2.4 Venn Diagrams with Three Sets and Verification of Equality of Sets 79 72. United States Blood Types The following table contains data from WorldAtlas.com regarding the percentage of people, in the United States, who have each blood type. Type Positive Blood, % Negative Blood, % O 38% 7% A 34% 6% B 9% 2% AB 3% 1% Construct a Venn diagram similar to the one in Example 2 and place the correct percentage in each of the eight regions. * 73. Categorizing Contracts J & C Mechanical Contractors wants to classify its projects. The contractors categorize set A as construction projects, set B as plumbing projects, and set C as projects with a budget greater than $300,000. a) Draw a Venn diagram that can be used to categorize the company projects according to the listed criteria. * b) Determine the region of the diagram that contains construction projects and plumbing projects with a budget greater than $300,000. Describe the region using sets A B, , and C with set operations. Use union, intersection, and complement as necessary. V; > > A B C c) Determine the region of the diagram that contains plumbing projects with a budget greater than $300,000 that are not construction projects. Describe the region using sets A B, , and C with set operations. Use union, intersection, and complement as necessary. VI; > > A B C ′ d) Determine the region of the diagram that contains construction projects and nonplumbing projects whose budget is less than or equal to $300,000. Describe the region using sets A B, , and C with set operations. Use union, intersection, and complement as necessary. I; > > A B C ′ ′ 74. Molecules Molecules that consist of carbon (C), hydrogen (H), and oxygen (O) are called carbohydrates. Molecules that consist of C and H only are called hydrocarbons. Molecules that consist of C and O only are called oxocarbons. Molecules that consist of H and O only are called oxohydrogens. Construct a Venn diagram in which the three sets are labeled C H, , and O. Label the 8 regions of the Venn diagram using the following: pure C molecules, pure H molecules, pure O molecules, carbohydrates, hydrocarbons, oxocarbons, oxohydrogens, and molecules that do not contain C, H, or O. * Challenge Problem/Group Activity 75. Define each of the eight regions in Fig. 2.25 using sets A B, , and C and a set operation. (Hint: > > A B C ′ ′ defines region I.) I > > A B C ′ ′ II > > A B C′ III > > A B C ′ ′ IV > > A B C ′ V > > A B C VI > > A B C ′ VII > > A B C ′ ′ VIII > > A B C ′ ′ ′ 76. We were able to determine the number of elements in the union of two sets with the formula < > nAB nAnBnAB ( ) ( ) ( ) ( ). = + − Can you determine a formula for determining the number of elements in the union of three sets? In other words, write a formula to determine < < n A B C ( ). [Hint: The formula will contain each of the following: n A n B n C ( ), (), ( ), > > > > > > nABCnAB CnA BC ( ), ( ), ( ), ′ ′ ′ and > > n A B C 2 ( ). * 77. a) Construct a Venn diagram illustrating four sets, A B C , , ,and D. (Hint: Four circles cannot be used, and you should end up with 16 distinct regions.) * b) Label each region with a set statement (see Exercise 75). Check all 16 regions to make sure that each is distinct. * Research Activity 78. Combining Colors The following two Venn diagrams illustrate what happens when colors are added or subtracted. Do research and write a report explaining the creation of the colors in the Venn diagrams, using such terms as union of colors and subtraction (or difference) of colors. U A B III II I IV V VI VII VIII C Figure 2.25 Yellow Magenta Blue Subtractive color mixing Green Red Additive color mixing Blueviolet *See Instructor Answer Appendix
RkJQdWJsaXNoZXIy NjM5ODQ=