Chapter 1 Review Math 1030Section C.1: Sets and Venn DiagramsDefinition of a setA set is a collection of objects and its objects are called members. For example, the “days of the week” is a setand Monday is a member of this set.Venn diagrams Venn diagrams are pictures that describe the relationships between sets. Circles representsets.Types 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.Ex.1 Draw a Venn diagram that represents the sets “men” and “women”.Ex.2 Draw a Venn diagram that represents the sets “female” and “children”.1Chapter 1 Review Math 1030Ex.3 Draw a Venn diagram that represents the sets “apples” and “fruits”.Section C.2: Categorical PropositionsDefinition of a propositionA proposition is a sentence that makes a claim. For example,• 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.Types of categorical propositionsThere are four types of categorical propositions:(a) All S are P.(b) No S are P.(c) Some S are P.(d) Some S are not P.If 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”.2Chapter 1 Review Math 1030Ex.4 Draw the Venn diagrams for: Some students are not female.Ex.5 Draw the Venn diagrams for: No cats are birds.Ex.6 Draw the Venn diagrams for: All children are happy.3Chapter 1 Review Math 1030Ex.7 Draw the Venn diagrams for: Some students have a job.Section C.3: Venn Diagrams with Three SetsEx.8Draw the Venn diagram for these sets: women, teachers, mothers.4Chapter 1 Review Math 1030Section C.4: More Uses of Venn DiagramsEx.9The following two-way table shows the number of men and women attending an orientation meeting, allof whom are undergraduate students math or physics majors and none is both. Fill in the missing figuresin the table and make a Venn diagram to represent the data.Men Women TotalMath 30 35Physics 15Total 1005Chapter 1 Review Math 1030Section D.1: Two Types of Arguments: Inductive and DeductiveDefinition of an argumentAn argument is a reasoned process that uses a set of facts or assumptions, called premises, to support aconclusion.There are two types of arguments: inductive and deductive.Definition of an inductive argumentAn inductive argument makes a case for a general conclusion from more specific premises.Ex.10 Inductive argument.Premise: Birds fly up into the air but eventually come back down.Premise: People who jump into the air fall back down.Premise: Rocks thrown into the air come back down.Premise: Balls thrown into the air come back down.Conclusion: What goes up must come down.Definition of a deductive argumentA deductive argument makes a case for a specific conclusion from more general premises.Ex.11 Deductive argument.Premise: All politicians are married.Premise: Senator Harris is a politician.Conclusion: Senator Harris is married.Evaluating Inductive ArgumentHow evaluate an inductive argumentAn inductive argument can be analyzed only in terms of its strength. An argument is strong if it makes acompelling case for its conclusion. It is weak if its conclusion is not well supported by its premises.An inductive argument cannot prove that its conclusion is true. At best, a strong inductive argument showsthat its conclusion is probably true.Ex.12Premise: Socrates was Greek.Premise: Most Greeks eat fish.Conclusion: Socrates ate fish.Evaluate this inductive argument.6Chapter 1 Review Math 1030Ex.13Premise: Erika loves cats.Premise: Michelle likes cats.Premise: Sharon likes cats.Conclusion: All women like cats.Evaluate this inductive argument.Evaluating Deductive ArgumentHow evaluate a deductive argumentA deductive argument can be analyzed in terms of its validity and soundness. An argument is valid if itsconclusion follows necessarily from its premises. It is sound if it is valid and its premises are true.Validity is concerned only with the logical structure of the argument: validity involves no personal judg-ment and has nothing to do with the truth of the premises or conclusion. Thus a deductive argument canbe valid even if its conclusion is false.A sound deductive argument provides definite proof that its conclusion is true. However, evaluatingsoundness often involves personal judgment.Tests of validityWe can test validity by examining the structure of a deductive argument in a systematic way using Venndiagrams:• Draw a Venn diagram that represents all the information contained in the premises.• Check to see whether the Venn diagram also contains the conclusion. If it does, then the argumentis VALID. If it doesn’t, then the argument is NOT VALID.Ex.14Premise: All men are mortal.Premise: Robert is a man.Conclusion: Robert is mortal.Evaluate this deductive argument.7Chapter 1 Review Math 1030Ex.15Premise: All women are mothers.Premise: Erika is a woman.Conclusion: Erika is a mother.Evaluate this deductive argument.Conditional Deductive ArgumentDefinition of a conditional deductive argumentA conditional deductive argument is a deductive argument in which the first premise is a conditional statement“if p, then q”.Types of conditional deductive argumentsThere are four basic types of conditional deductive arguments:(a) Affirming the HypothesisPremise: If p, then q.Premise: p is true.Conclusion: q is true.(b) Affirming the ConclusionPremise: If p, then q.Premise: q is true.Conclusion: p is true.(c) Denying the HypothesisPremise: If p, then q.Premise: p is not true.Conclusion: q is not true.(d) Denying the ConclusionPremise: If p, then q.Premise: q is not true.Conclusion: p is not true.See Homework problems 55 − 58 for examples.8Chapter 1 Review Math 1030Deductive Argument with a Chain of ConditionalsDefinition of a chain of conditionalsA deductive argument with a chain of conditionals is a deductive argument with the premises given byconditionals.• VALID chain of conditionals:Premise: If p, then q.Premise: If q, then r.Conclusion: If p, then r.• INVALID chain of conditionals:Premise: If p, then q.Premise: If r, then q.Conclusion: If p, then r.Ex.16Premise: If