1. Direct Proof and Counterexample II
Lecture 14
Section 3.2
Fri, Feb 9, 2007

2. Rational Numbers
A rational number is a number that equals the quotient of two integers.
An irrational number is a number that is not rational.

3. Direct Proof Theorem: The sum of two rational numbers is rational.
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
Prove or disprove: The sum of two irrationals is irrational.

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

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:
Let a and b be odd integers.
Then a = 2s + 1 and b = 2t + 1 for some integers s and t.

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

9. Direct Proof
Theorem: If you start with the number of the month in which you were born, multiply it by 4, add 13, multiply the result by 25, subtract 200, add the day of the month on which you were born, multiply by 2, subtract 40, multiply the result by 50, add the last two digits of the year of your birth, and finally, subtract 10,500, you will get…

10. Other Theorems
Theorem: Between every two distinct rationals there is a rational.
Proof: ?
Theorem: Between every two distinct irrationals there is a rational.
Theorem: Between every two distinct irrationals there is an irrational.

11. An Interesting Question
Which of the previous theorems are the hardest to prove?
Why?

12. An Interesting Question
Because they involve “negative” hypotheses and “negative” conclusions.

13. 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.

14. 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.

15. Proving Positive Statements
Theorem: Let P be a point on the line
x + y = 4
and let Q be a point on the circle
x2 + y2 = 1.
Then the distance from P to Q is at least 1.
Proof: