Chapter 2

Section 2.2

Applications of Sets

References to Various parts of a Venn DiagramThe 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.

ExampleA 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 turtlesH = Set of heavy turtles (over 2kg)F H

U

38

10

9 271. 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 38+9+27=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?

910

2774

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.

ExampleAt 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.

TPU

18

22392

1. There are 18 that are neither parents nor teachers.2. That leaves 135-18=117 parents or teachers.3. Now 117-95=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?

110

3

117

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 (').

UA B

C

CBA This is what is in all three sets A and B and C.

UA B

C

'CBA This is what is in the two sets A and B but not in set C.

UA B

C

'' CBA 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 research133 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

2529

13

21

28 17

58 22

P M

E

2529

13

21

28 17

58 22

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 21.

2. How many are in the set ? In other words how many gave to neither politics nor medicine?

It would be 58+22, which is 80.

3. How many gave only to medicine? (In other words )

It would be 17.

EMP '

)'( MP

'' EMP