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Direct Proof and Counterexample IIRational NumbersDirect ProofProof by CounterexampleSlide 5Slide 6Proof, continuedOther TheoremsAn Interesting QuestionPositive and Negative StatementsSlide 11Irrational NumbersDirect Proof and Counterexample IILecture 13Section 3.2Wed, Feb 2, 2005Rational NumbersA rational number is a number that equals the quotient of two integers.Let Q denote the set of rational numbers.An irrational number is a number that is not rational.We will assume that there exist irrational numbers.Direct ProofTheorem: The sum of two rational numbers is rational.Proof:Let r and s be rational numbers.Let r = a/b and s = c/d, where a, b, c, d are integers.Then r + s = (ad + bc)/bd, which is rational.Proof by CounterexampleDisprove: The sum of two irrationals is irrational.Counterexample:Proof by CounterexampleDisprove: The sum of two irrationals is irrational.Counterexample:Let α be irrational.Then -α is irrational. (proof?)α + (-α) = 0, which is rational.Direct ProofTheorem: Between every two distinct rationals, there is a rational.Proof:Let r, s Q. WOLOG* WMA† r < s.Let t = (r + s)/2.Then t Q. (proof?)We must show that r < t < s.*WOLOG = Without loss of generality†WMA = We may assumeProof, continuedGiven that r < s, it follows that 2r < r + s < 2s.Then divide by 2 to get r < (r + s)/2 < s.Therefore, r < t < s.Other TheoremsTheorem: Between every two distinct irrationals there is a rational.Proof: Difficult.Theorem: Between every two distinct irrationals there is an irrational.Proof: Difficult.An Interesting QuestionWhy are the last two theorems so hard to prove?Because they involve “negative” hypotheses and “negative” conclusions.Positive and Negative StatementsA positive statement asserts the existence of a number.A negative statement asserts the nonexistence of a number.It is much easier to use a positive hypothesis than a negative hypothesis.It is much easier to prove a positive conclusion than a negative conclusion.Positive and Negative Statements“r is rational” is a positive statement.It asserts the existence of integers a and b such that r = a/b.“α is irrational” is a negative statement.It asserts the nonexistence of integers a and b such that α = a/b.Is there a “positive” characterization of irrational numbers?Irrational NumbersTheorem: Let be a real number and define the two setsA = iPart({1, 2, 3, …}*( + 1))andB = iPart({1, 2, 3, …}*(-1 + 1)).Then is irrational if and only if A B = N and A B =

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