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U of U MATH 1030 - Thinking Critically Lecture notes

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Chapter 1: Thinking Critically Lecture notes Math 1030 Section CSection C.1: Sets and Venn DiagramsDefinition of a setA set is a collection of objects and its objects are called members.Ex.1 Days of the week.Ex.2 Students in this class.Ex.3 Letters of the alphabet.Venn diagrams Venn diagrams are pictures that describe the relationships between sets. Circles representsets.Ex.4 The set of “students”.1Chapter 1: Thinking Critically Lecture notes Math 1030 Section CEx.5 All the students in this class are students.We say that the “students in this class”, as a set, is a subset of the set “students”.Ex.6 If we consider the sets “dogs” and “cats”, then there is not a member which is both a dog and acat.We say that “dogs” and “cats” are disjoint sets.Ex.7 Consider the sets “men” and “students”.We say that “men” and “students” are overlapping sets.2Chapter 1: Thinking Critically Lecture notes Math 1030 Section CTypes of Venn diagrams for two setsThe Venn diagrams for two sets A and B are of three types:(a) A is a subset of B, which means that all the members of A are members of B.(b) A is disjoint from B, which means that A and B doesn’t have any member in common.(c) A and B are overlapping sets, which means that they (may) share some of the members.3Chapter 1: Thinking Critically Lecture notes Math 1030 Section CEx.8 Numbers.• Set of natural numbers:• Set of whole numbers:• Set of integers:• Set of rational numbers:the rational numbers are of the formxy, where x and y are integers and y 6= 0.For example,12,35,94.When expressed in decimal forms, rational numbers are either terminating decimals with a finitenumber of digits (for example,14= 0.25) or repeating decimals in which a pattern repeats over andover (for example,13= 0.333 . . . = 0.¯3).• Set of irrational numbers:the irrational numbers are numbers that cannot be expressed in the formxy, where x and y areintegers and y 6= 0. When written as decimals, irrational numbers neither terminate nor have arepeating pattern.For example, π,√2:π = 3.14159 . . . ,√2 = 1.4142 . . . .• Set of real numbers:the set of real numbers consists of both rational and irrational numbers.The Venn diagram representing the relationship between these sets of numbers is:4Chapter 1: Thinking Critically Lecture notes Math 1030 Section CSection C.2: Categorical PropositionsDefinition of a propositionA proposition is a sentence that makes a claim.Ex.9 Examples of propositions:• Today it is raining.• All whales are mammals.Definition of a categorical propositionA categorical proposition is a proposition which claims a particular relationship between two categories orsets.5Chapter 1: Thinking Critically Lecture notes Math 1030 Section CTypes of categorical propositionsThere are four types of categorical propositions:(a) All whales are mammals.(b) No fish are mammals.(c) Some doctors are women.(d) Some teachers are not men.6Chapter 1: Thinking Critically Lecture notes Math 1030 Section CIf a categorical proposition is not in the standard form like “Some birds can fly”, we can change it as follows:“Some birds are animals that can fly”.Ex.10 How can we change “Elephants never forget” into the standard form?Ex.11 How can we change “Every nurse knows CPR” into the standard form?7Chapter 1: Thinking Critically Lecture notes Math 1030 Section CSection C.3: Venn Diagrams with Three SetsEx.12 Students at the U.Consider the students at the U and the sentences:• Some of the honor-student boys are unemployed.• Some of the non-honor-student girls are employed.How can we draw a picture representing this situation?Ex.13 Problem 63.Draw the Venn diagram for these sets: women, dentists, kindergarten teachers.8Chapter 1: Thinking Critically Lecture notes Math 1030 Section CEx.14 Problem 65.Draw the Venn diagram for these sets: published works, novels, songs.9Chapter 1: Thinking Critically Lecture notes Math 1030 Section CSection C.4: More Uses of Venn DiagramsEx.15 Application of Venn diagrams.Consider the two-way table:Baby’s Birth Weight StatusLow Birth Weight Normal Birth WeightMother’s Smoking StatusSmoker 18 132Nonsmoker 14 186We can use a Venn diagram to represent the situation:10Chapter 1: Thinking Critically Lecture notes Math 1030 Section CEx.16 Problem 69: A Venn diagram with numbers.(1) How many men are in the room?(2) How many non-Republicans are in the room?(3) How many Republican women are in the room?(4) How many people are in the room?11Chapter 1: Thinking Critically Lecture notes Math 1030 Section CEx.17 Problem 87: Completing a two-way table.The following two-way table shows the number of patients in a study who responded to either a newallergy drug or a placebo (a look-like pill with no active ingredients). A total of 100 patients participated inthe study. Fill in the missing figures in the table and make a Venn diagram to represent the data.Showed improvement No improvement TotalNew drug 20 35Placebo 40Total 10012Chapter 1: Thinking Critically Lecture notes Math 1030 Section CEx.18Of the 100 students in the Science Club, 40 are taking a physics class, 42 are taking a chemistry class, and15 are taking a math class. Moreover, 30 students are taking chemistry only, 24 students are taking physicsonly, 6 students are taking math only, 5 students are taking physics and math only, 8 students are takingphysics and chemistry only. Draw a Venn diagram to illustrate this information. Use the symbol P, C, Mto represent the set of physics, chemistry, and math students respectively.Use your diagram to answer the following:(1) How many students are taking all three subjects?(2) How many students in the Science Club are not taking any of these three


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