Direct Proof and Counterexample II

1. Direct Proof and Counterexample II Lecture 12 Section 3.2 Thu, Feb 9, 2006

2. Rational Numbers • A 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.

3. Direct Proof • Theorem: 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, where b, d > 0. • Then r + s = (ad + bc)/bd.

4. Direct Proof • We know that ad + bc is an integer. • We know that bd is an integer. • We also know that bd 0. • Therefore, r + s is a rational number.

5. Proof by Counterexample • Disprove: The sum of two irrationals is irrational. • Counterexample:

6. Proof by Counterexample • Disprove: The sum of two irrationals is irrational. • Counterexample: • Let α be irrational. • Then -α is irrational. (proof?) • α + (-α) = 0, which is rational.

7. Direct Proof • Theorem: The sum of two odd integers is an even integer; the product of two odd integers is an odd integer. • Proof:

8. Direct Proof • Theorem: The sum of two odd integers is an even integer; the product of two odd integers is an odd integer. • Proof: • Let a and b be odd integers. • Then a = 2s + 1 and b = 2t + 1 for some integers s and t.

9. Direct Proof • Then a + b = (2s + 1) + (2t + 1) = 2(s + t + 1). • Therefore, a + b is an even integer. • Finish the proof.

10. Direct Proof • Theorem: 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?) *WOLOG = Without loss of generality †WMA = We may assume

11. Proof, continued • We must show that r < t < s. • Since r < s, it follows that 2r < r + s < 2s. • Then divide by 2 to get r < (r + s)/2 < s. • Therefore, r < t < s.

12. Other Theorems • Theorem: Between every two distinct irrationals there is a rational. • Proof: Difficult. • Theorem: Between every two distinct irrationals there is an irrational. • Proof: Difficult.

13. An Interesting Question • Why are the last two theorems so hard to prove? • Because they involve “negative” hypotheses and “negative” conclusions.

14. Positive and Negative Statements • A 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.

15. 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.

16. Positive and Negative Statements • Is there a “positive” characterization of irrational numbers?

17. Irrational Numbers • Theorem: Let  be a real number  and define the two sets A = iPart({1, 2, 3, …}( + 1)) and B = iPart({1, 2, 3, …}(-1 + 1)). Then  is irrational if and only if AB = N and A B = . • Try it out: Irrational.exe