copyright © 2015, 2011, 2008 pearson education, inc. chapter 1, unit 1c, slide 1 thinking...

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1C, Slide 1

ThinkingCritically1


Unit 1C

Sets and Venn Diagrams


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.


ExampleUse braces to write the contents of each of the following sets:the set of countries larger in land area than the United Statesthe 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.


The set whales is a subset of the set mammals.

Venn Diagrams


The set of fish is disjoint from the set mammals.

other animals

Venn Diagrams


men who are not doctors

The sets of doctors and women are overlapping.

male doctors

female doctors

women who are not doctors

Venn Diagrams


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.


ExampleDescribe 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


ExampleDescribe 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.


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.)


Real Number Venn Diagram


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.


All S are P

No S are P

All whales are mammals

No fish are mammals

Categorical Propositions


Some S are P

Some S are not P

Some doctors are women

Some teachers are not men

Categorical Propositions


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


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.


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.


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.


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.


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.


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.


Blood Types

Venn Diagrams with Three Sets

copyright © 2015, 2011, 2008 pearson education, inc. chapter 1, unit 1c, slide 1 thinking...

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