Thinking Critically

1. 1 Thinking Critically Sets and Venn Diagrams

2. Definitions • A set is a collection of objects. • The members of a set are the individual objects within it. • Write sets by listing their members within a pair of braces, { }. • Use three dots, …, to indicate a continuing pattern if there are too many members to list. • A Venn diagram is a diagram that uses circles to represent sets.

3. Example Use braces to write the contents of each of the following sets: • the set of countries larger in land area than the United States • the set of natural numbers greater than 5 Solution The set of countries larger in land area than the United States is {Russia, Canada}. The set of natural numbers greater than 5 is {6, 7, 8, . . .}; the dots indicate that the list continues to ever-higher numbers.

4. Venn Diagrams The set whales is a subset of the set mammals.

5. Venn Diagrams other animals The set of fish is disjoint from the set mammals.

6. Venn Diagrams men who are not doctors male doctors female doctors women who are not doctors The sets of doctors and women are overlapping.

7. Set Relationships • If A is a subset of B, then all members of A are also members of B. • If A is disjoint from B, then the two sets have no members in common. • If A and B are overlapping sets, then the two sets share some of the same members.

8. Example Describe the relationship between Democrats and Republicans (party affiliations), and draw a Venn diagram showing this relationship. Interpret all the regions of the Venn diagram. Solution

9. Example Describe the relationship between Democrats and Republicans (party affiliations), and draw a Venn diagram showing this relationship. Interpret all the regions of the Venn diagram. Solution A person can be registered for only one political party, so the sets Democrats and Republicans are disjoint. The region outside both circles represents people who are neither Democrats nor Republicans—that is, people who are registered for other political parties, who are independent, or who are not registered.

10. Example Draw a Venn diagram showing the relationships among the sets of natural numbers, whole numbers, integers, rational numbers, and real numbers. Where are irrational numbers found in this diagram? (If you’ve forgotten the meanings of these number sets, see Brief Review on p. 27.)

11. Real Numbers • Rational numbers are all numbers that can be represented as a fraction (irrational numbers cannot be represented as a fraction). • Integers are whole numbers and their negatives or opposites.

12. Categorical Propositions • Categorical propositions must have the structure of a complete sentence. • One set appears in the subject. • The other appears in the predicate. • For example, in the proposition all whales are mammals, the set whales is the subject set and the set mammals is the predicate set. We usually use the letter S to represent the subject set and P for the predicate set, so we can rewrite all whales are mammals as all S are P, where S = whales and P = mammals.

13. Categorical Propositions All whales are mammals No fish are mammals All S are P No S are P

14. Categorical Propositions Some doctors are women Some teachers are not men Some S are P Some S are not P

15. Constructing a Venn Diagram Statement: “Some dogs can swim.” Rephrase to: “Some dogs are animals that can swim.” Construct the Venn diagram: S = dogs P = animals that can swim

16. Example Consider the study summarized in Table 1.1, which is an example of a two-way table. This study was designed to learn whether a pregnant mother’s status as a smoker or nonsmoker affects whether she delivers a low or normal birth weight baby. The table shows four numbers, which correspond to the four possible combinations of the baby’s birth weight status and the mother’s smoking status.

17. Example (cont) a. Make a list summarizing the four key facts shown in the table. b. Draw a Venn diagram to represent the table data. c. Based on the Venn diagram, briefly summarize the results of the study.

18. Example (cont) a. Make a list summarizing the four key facts shown in the table. • 18 babies were born with low birth weight to smoking mothers. • 132 babies were born with normal birth weight to smoking mothers. • 14 babies were born with low birth weight to nonsmoking mothers. • 186 babies were born with normal birth weight to nonsmoking mothers.

19. Example (cont) b. Draw a Venn diagram to represent the table data. The figure shows one way of making the Venn diagram. The circles represent the sets smoking mothers and low birth weight babies. The labels show how each region correspondsto one of the entries in Table 1.1.

20. Example (cont) The Venn diagram makes it easy to see how smoking affected babies in the study. Notice that normal birth weight babies were much more common than low birth weight babies among both smokers and nonsmokers. However, the smoking mothers had a lower proportion of normal birth weight babies and a higher proportion of low birth weight babies. This suggests that smoking increases the risk of having a low birth weight baby, a fact that has been borne out by careful statistical analysis of this and other studies.

21. Example Human blood is often classified according to whether three antigens, A, B, and Rh, are present or absent. Blood type is stated first in terms of the antigens A and B: Blood containing only A is called type A, blood containing only B is called type B, blood containing both A and B is called type AB, and blood containing neither A nor B is called type O. The presence or absence of Rh is indicated by adding the word positive (present) or negative (absent) or its symbol. Table 1.2 (next slide) shows the eight blood types that result and the percentage of people with each type in the U.S. population. Draw a Venn diagram to illustrate these data.

22. Venn Diagrams with Three Sets Blood Types