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Section 1.4: Sets and Venn Diagrams: Pictorial Logic

A set is a collection of objects, which are called elements of the set.

Example 1: If V is the collection of all vowels in the alphabet, specify the set as a list of elements.

The set B is a subset of the set A if every element of B is also an element of A.

Two sets are called disjoint if they have no elements in common.

Example 2: Let S be the set of U.S. senators, R the set of members of the U.S. House of Representatives, and G the set of all elected U.S. government officials.

a. How many elements are in the set S? b. Of the three sets, which are subsets of others? c. Which pairs of the three sets S, R, and G are disjoint?

A Venn diagram is a visual device for representing sets. It usually consists of circular regions inside a rectangle.

Example 3: Let A = {2, 3, 4, 5, 6} and B = {4, 5, 6, 7, 8}. Make a Venn diagram showing elements in each set.

Example 4: Medical tests do not always produce accurate results. For this example, we use the category of all people who have undergone a medical test for hepatitis C.

Let H denote the set of all patients who have hepatitis C, and let P be the set of all people who test positive.

Make a Venn diagram using these sets, and locate the following regions:

• True positives: These are people who test positive and have hepatitis C. • False positives: These are people who test positive but do not have hepatitis C. • True negatives: These are people who test negative and do not have hepatitis C. • False negatives: These are people who test negative but do have hepatitis C.

Example 5: A double-blind drug trial separated a group of 100 patients into a test group of 50 and a control group of 50. The test group got the experimental drug, and the control group got a placebo. The result was that 40% of the test group improved and only 20% of the control group improved.

Let T be the test group and I the set of patients who improved.

Make a Venn diagram that shows the number of people in each of the following cate- gories:

• Test group who improved and Test group who did not improve • Control group who improved Control group who did not improve

Logic and Venn diagrams:

Example 6: Consider the conditional statement:

If you are stopped for speeding, then you will get a traffic ticket.

Let S = {People stopped for speeding} and T = {People who get traffic tickets}.

Example 7: (Conjunction using Venn diagrams.) For the conjunction Joe is a math major and a music major,

Let A = {Math majors} and B = {Music majors}

To say Joe is a math and a music major means that he belongs to both sets (Fig. 1.12).

Example 8: (Disjunction using Venn diagrams.) For the disjunction Joe is a math major or a music major,

Let A = {Math majors} and B = {Music majors}

To say Joe is a math or a music major means that Joe belongs to one set or the other (Fig. 1.13).

Example 9: Let F be the set of senators who voted for the farm bill and T the set of senators who voted for the tax bill.

1. Make a Venn diagram for F and T , and shade in the region for which the state- ment He voted for the tax bill but not the farm bill is true.

2. Let S denote the set of all Southern senators. Suppose it is true that all South- ern senators voted for both the tax and farm bills. Add the circle representing Southern senators to the Venn diagram you made in part 1.

Counting using Venn diagrams

Example 10: Suppose that 17 tourists travel to Mexico or Costa Rica. Some visit Mexico, some visit Costa Rica, and some visit both. If 10 visit Mexico but not Costa Rica, and 5 visit both, how many people visit Costa Rica?

M = {Tourists visiting Mexico} C = {Tourists visiting Costa Rica}

Example 11: Suppose that 100 high school students are examined in science and mathematics. Here are the results.

• 41 students passed the math exam but not the science exam. • 19 students passed both exams. • 8 students failed both exams.

How many students passed the math exam, and how many passed the science exam?

Venn diagrams with three sets

In the Venn diagram in Figure 1.20, the rectangle represents all highway collisions. The three circles show the percentage of collisions resulting from one or a combination of three factors: the vehicle, human error, and the road.

Example 12: A survey asked 110 high school athletes which sports they played among football, basketball, and baseball. The results are given in the table below.

Football only 24

Basketball only 13

Baseball only 12

Football and baseball, but not basketball 9

Football and basketball, but not baseball 5

Basketball and baseball, but not football 10

All three sports 6

How many played none of these three sports?

How many played exactly two of the three sports?

Practice Problems from Exercise Set 1.4: 1, 2, 5–7, 11, 12–19, 25–28, 31, 40–44