Chapter 2 Section 2.2 Applications of Sets

References to Various parts of a Venn Diagram The information that is told to you might not always correspond directly to the center portion of the Venn Diagram. It might be one of the other parts of the diagram. You need to read carefully to see exactly which part of the diagram the problem is referring to. Example A biologist captured 84 snapping turtles. There were 47 that were female and 36 that weighed more than 2 kg or more. There were 38 female turtles that weighed less than 2kg. Fill in a Venn Diagram for this. F = Set of female turtles H = Set of heavy turtles (over 2kg) F H U There were 38 females that were not heavy. 2. That leaves 9 to make up the 47 females. 3. Need another 27 to have 36 that are heavy. 4. Now =74 need 10 more to get 84 total. a) How many are both female and heavy? b) How many are neither female nor heavy? c) How many are heavy male turtles? d) How many are either female or heavy?

In some problems the number that is outside both of the categories (circles in the Venn Diagram) will be told to you rather than one of the values in the center. Example At a recent school board meeting the were 135 people sitting in the audience. A survey showed that 95 were a parent of a child in the school system and 25 were a teacher in the school system. There were 18 that were neither a parent nor a teacher in the system. Fill in a Venn Diagram for this. P = Set of parents in the audience. T = Set of teachers in the audience. T P U There are 18 that are neither parents nor teachers. 2. That leaves =117 parents or teachers. 3. Now =22 that are only teachers. 4. That leaves 3 to make up the 25 teachers. 5. That leaves another 92 to make up the 95 parents. a)How many are not teachers? b)How many are both parents and teachers? c)How many are either parents or teachers?

Dealing with 3 Sets We can handle three categories at once in a similar way that we handled two categories. The Venn Diagram is slightly changed, but all of the regions can be referred to using the set operations union (  ), intersection (  ) and complement ('). U A B C This is what is in all three sets A and B and C. U A B C This is what is in the two sets A and B but not in set C. U A B C This is what is not in the sets A and what is in the set B but not in set C.

Relations With 3 Sets The problems that we were working on relating two sets can also be done with three sets. This is very useful for understanding survey information for example. Problem: In a recent survey of monetary donations made by college graduates, the following information was obtained: 95 donated to a political campaign 76 donated to medical research 133 donated to the environment 25 donated to all three 22 donated to none 38 donated to politics and medicine 46 donated to medicine and environment 54 donated to a politics and environment We begin by naming three sets: P = People who made political donations M = People who made medical donations E = People who donated to the environment The idea is to start from the very center and work your way outward. P M E

P M E Now that we know how the categories break down more detailed questions can be asked about this situation. This is a matter of pulling the information out of the correct part of the Venn Diagram. Look at each question below. 1. How many gave to medicine ( M ) and the environment ( E ) but not politics ( P )? (In other words ) It would be How many are in the set ? In other words how many gave to neither politics nor medicine? It would be 58+22, which is How many gave only to medicine? (In other words) It would be 17.