Section 1.4: Applications of Set Theory Set theory is useful for solving many types of problems, including Internet searches, database queries, data analyses, games, and puzzles. Analyzing 3 intersecting sets A B C is represented by region h, A B is represented by the union of regions e and h, A C is represented by the union of regions g and h, and B C is represented by the union of regions h and i. Each region of a Venn diagram contains elements that occur only in that particular region. You can use a Venn diagram or the Principle of Inclusion and Exclusion to determine the number of elements in the union of three sets: n(a B C) = n(a ) + n(b) +n(c) n(a B) n(a C) n(b C) + n(a B C) 1
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Examples: 1. A group of 30 students are surveyed to fi nd out which of the three sports, soccer (S), basketball (B) or volleyball(v), they play. The results are as follows: 3 children do not play any of these sports 2 children play all three sports 6 play volleyball and basketball 3 play soccer and basketball 6 play soccer and volleyball 16 play basketball 12 play volleyball (i) How many students play soccer only? (ii) How many students play soccer but not basketball? (iii) How many students play volleyball but not basketball? Start by completing the centre, then the two set intersections before filling the remainder of each set. 3
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2. There are 36 students who study science. 14 study physics, 18 study chemistry, 24 study Biology, 5 study physics and chemistry, 8 study physics and biology, 10 study biology and chemistry, 3 study all three subjects. (i) Determine the number of students who study Physics and Biology only. (ii) Determine the number of students who study at least two subjects. (iii) Determine the number of students who study biology only. 5
3. A survey of a machine shop reveals the following information about its employees: 44 employees can run a lathe, 49 employees can run the milling machine, 56 employees can operate a punch press, 27 employees can run a lathe and a milling machine, 19 employees can run a milling machine and operate a punch press, 24 employees can run a lathe and operate a press punch, 10 employees can operate all three machines, 9 employees cannot operate any of the three machines. Ask students to determine the number of people employed at the machine shop. 6
4. A firm manufactures three types of shampoo: Shine, Bubble, and Glory. 1000 families were surveyed and the results were posted in an advertisement as follows: 843 use Shine 673 use Bubble 585 use Glory 600 use both Shine and Bubble 423 use both Shine and Glory 322 use both Bubble and Glory 265 use all three types Identify where the error occurs in the survey results as reported in the advertisement. 7
There are problems involving sets where you will have to use an equation to solve the problem. You will have to use the clues or information to define the sets, identify regions of the Venn diagram, and identify the region of interest using a variable. 5. There are twenty five dogs at the dog show. Twelve of the dogs are black, eight of the dogs have short tails, and fifteen of the dogs have long hair. There is only one dog that is black with a short tail and long hair. Three of the dogs are black with short tails and do not have long hair. Two of the dogs have short tails and long hair but are not black. If all of the dogs in the kennel have at least one of the mentioned characteristics, how many dogs are black with long hair but do not have short tails? 8
6. 40 members in a sports club were surveyed: 2 play all three sports 23 play ball hockey 24 play tennis 18 play golf 14 play tennis and ball hockey 8 play tennis and golf 1 member makes the refreshments and does not play any sport Determine the number of people who play ball hockey and golf. 9
7. In a survey of 55 people, the following results were recorded: 13 people like Hawaiian pizza 19 people like pepperoni pizza 26 people like cheese pizza 15 people do not like pizza 5 people like Hawaiian pizza and pepperoni pizza, but not cheese pizza 2 people likes all types of pizza 2 people like Hawaiian pizza and cheese pizza, but not pepperoni pizza Determine how many people like only cheese pizza. 10