1549901369-Elements_of_Real_Analysis__Denlinger_

A.l The Logic of Propositions 591 We can use truth-tables to verify (prove) that these are tautologies. For example, to verify ( ...

592 Appendix A • Logic and Proofs regardless of the truth-values of P and Q. By Definition A.l.16, this means that these compoun ...

A. l The Logic of Propositions 593 CONVERSE, INVERSE AND CONTRAPOSITIVE Associated with a given implication, P ::::} Q, there a ...

594 Appendix A • Logic and Proofs Table A.10 p Q R QVR p /\ p /\ Q p /\ R (P /\ Q) V (Q V R) (P /\ R) T T T T T T T T T T F T T ...

A.2 The Logic of Predicates and Quantifiers 595 In Exercises 21 - 28 state the negation of the given symbolic statement, and use ...

596 Appendix A • Logic and Proofs Such statements are sometimes called "open sentences" -the variables re- serve open spaces in ...

A.2 The Logic of Predicat es and Quantifiers 597 symbolized P(x, y), Q( u, v), etc. Of course, we can have a propositional funct ...

598 Appendix A • Logic and Proofs Examples A.2.6 Express each of the following English sentences in symbolic form: (a) Everyone ...

A.2 The Logic of Predicates and Quantifiers 599 This is read "there exists an x such that P( x)." The symbol ":3 x" is called th ...

600 Appendix A • Logic and Proofs ( d) Let the domain of x be the set of all real numbers. The given statement is ,...., 3 x 3 x ...

A.2 The Logic of Predicates and Quantifiers 601 (f) Somebody stole my wallet. (g) The equation x^2 - x - 6 = 0 has a solution in ...

602 Appendix A • Logic and Proofs having modern logical symbolism available, logicians expressed these proposi- tions in words: ...

A.2 The Logic of Predicates and Quantifiers 603 Theorem A.2.15 Negations of Categorical Propositions: (a) '""UP =EN (b) '""EP =U ...

604 Appendix A • Logic and Proofs the original statement. Incorrect quantification can completely destroy the in- tended meaning ...

A.2 The Logic of Predicates and Quantifiers 605 No one in this class will fail if he does the assignments regularly. Someone wa ...

606 Appendix A • Logic and Proofs PART B: Write the negation of each proposition in Part A, first in sym- bols, and then in word ...

A.3 Strat egies of Proving Theorems 607 We shall find it co nvenient to refer to this schematic in horizontal form: Table A.12 T ...

608 Appendix A • Logic and Proofs Proof. Use a truth table to show that [P =? Q and Q =? R] =? [P =? R] is a tautology. • Here's ...

A.3 Strategies of Proving Theorems 609 (PS-3') TO PROVE A THEOREM BY ITS CONTRAPOSITIVE: To prove a theorem: H,:.C, we prove the ...

610 Appendix A • Logic and P roofs Logicians call this tautology "constructive dilemma." It can be proved by a truth table. In w ...