Chapter 1 The Logic of Compound Statements    1

    1．1 LogicalForm and LogicalEquivalence   1

Statements；CompoundStatements；TruthValues；EvaluatingtheTruthofMo

re General Compound Statements；Logical Equivalence；Tautologies

and Contradictions；Summary ofLogical Equivalences

    1．2 Conditional Statements    17

        Logical Equivalences Involving→：Representation ofIf-Then

As Or；The Negadon of a Conditional Statement；The Contrapositive

of a Conditional Statement；The Converse and Inverse of a

Conditional Statement；Only If and the Biconditional；Necessary and

Sufficient Conditions；Remarks

    l.  3 Valid andInvalid Arguments    29

        Modus Ponens and Modus Tollens；Additional Valid Argument

Forms：Rules of

        Inference；Fallacies；Contradictions and Valid

Arguments；Summary of Rules of

        Inference

    1．4Application：Digital Logic Circuits 43

        Black Boxesand Gates；The Input／Output for a Circuit；The

Boolean Expression Cor-

        responding to a Circuit；The Circuit Corresponding to a

Boolean Expression；Finding

        a CircuitThatCorresponds to a

GivenInput／OutputTable；Simplifying Combinational

        Circuits；NAND and NOR Gares

    1．5 Application：Number Systems and Circuits for Addition    

57

        Binary Representation of Numbers；Binary Addition and

Subtraction；Circuits for

Computer Addition；Two"s Complements and the Computer

Representation of Neg-

        ativeIntegers；8-Bit Representation of a

Number；ComputerAddition with Negative

        Integers；Hexadecimal Notation

Chapter 2 The Logic of Quantified Statements    75

    2．1 Introduction to Predicates and Quantified Statements /    

75

        The Universal Quantifier：V：The Existential Quantifier：ョ

：Formal Versus Informal

        Language；Universal Conditional Statements；Equivalent

Forms ofthe Universal and

        Existential Statements；Implicit Quantification；Tarski"s

World

    2．2 Introduction to Predicates and Quantified Statements II 88

        Negations of Quantified Statements；Negations of Universal

Conditional Statements；The Relation among V，ョ，∧，and V；Vacuous

Truth of Universal Statements；Variants

        0f Universal Conditional Statements；Necessal-y and

Sufficient Conditions，Only If

2．3 Statements Containing Multiple Quantifiers    97

        Translating from Informal to Formal Language；Ambiguous

Language；Negations of Multiply．Quantified Statements；Older of

Quantifiers；Formal Logical Notation；Prolog

    2. 4 Arguments with Quantified Statements    111

       Universal MOdus Ponens；Use of Universal Modus Ponens in a

Proof；Universal Modus Tollens；proving Validity of Arguments with

Quantified Statements；Using Diagramsto

Test for Validity；Creating Additional Forms of Argument；Remark on

the Converse and Inverse Errors

Chapter 3 Elementary Number Theoryand Methods ofProof    125

     3.1 Direct Proofand Counterexample h Introduction    126

        Definitions；Provlag Existential Statements；Disproving

Universal Statements by

        Counterexample；Proving Universal Statements；Directions

for Writing Proofs of

        Universal Statements；Common Mistakes；Getting Proofs

Started；Showing That an

        Existential Statement Is False；Conjecture，Proof,and

Disproof

   3．2 Direct Proofand Counterexample II Rational Numbers    141

        More on Generalizing from the Generic Particular；Proving

Properties of Rational

        Numbers；Deriving New Mathematics from Old

    3．3 Direct Proof and Counterexample IIh Divisibility    148

        Pmving Properties of Divisibility；Counterexamples and

Divisibility；The Unique

        Factorization Theorem

    3．4 Direct Proof and Counterexample IV: Division into Cases

and the Quotient-Remainder Theorem    156

        Discussion of the Quorient．Remainder Theorem and Examples

；d／v and mod；Alter-

        native Representations of Integers and Applications to

Number Theory

    3．5 Direct Proofand Counte