1. Direct Proof and Counterexample I
Lecture 13
Section 3.1
Wed, Feb 7, 2007

2. Proving Universal Statements
A universal statement is generally of the form
x  D, P(x)  Q(x)
Use the method of generalizing from the generic particular.
Select an arbitrary x  D (generic particular).
Assume that P(x) is true (hypothesis).
Argue that Q(x) is true (conclusion).

3. Example: Direct Proof
Theorem: If n is an odd integer, then n3 – n is a multiple of 12.
Proof:
Let n be an odd integer.
Then n = 2k + 1 for some integer k.
Then n3 – n = (2k + 1)3 – (2k + 1)
= 8k3 + 12k2 +4k
= 4k(2k2 + 1) + 12k2.

4. Example: Direct Proof Now divide into cases: k is a multiple of 3:
Then we are done.
If k is not a multiple of 3,
Then k = 3m  1 for some integer m.
And 2k2 + 1 = 2(3m  1)2 + 1
= 18m2  12m +3
= 3(6m2  4m + 1).
Therefore, n3 – n is a multiple of 12.

5. An Alternate Proof Proof:
n3 – n = (n – 1)(n)(n + 1), which is the product of 3 consecutive integers.
One of them must be a multiple of 3.
Since n is odd, n – 1 and n + 1 must be even, i.e., multiples of 2.
Therefore, n3 – n must be a multiple of 12.

6. Example: Direct Proof Theorem: If x, y  R, then x2 + y2  2xy. Proof:
Let x, y  R.
x2 – 2xy + y2  0.
(x – y)2  0, which is known to be true.

7. Example: Direct Proof Theorem: If x, y  R, then x2 + y2  2xy. Proof:
Let x, y  R.
x2 – 2xy + y2  0.
(x – y)2  0, which is known to be true.
What is wrong with this “proof?”

8. Example: Direct Proof Correct proof: Let x, y  R. (x – y)2  0.
x2 – 2xy + y2  0.
x2 + y2  2xy.

9. Example: Disproving an Existential Statement
Theorem: There is no solution in real numbers to the equation
2x = x.

10. Example: Disproving an Existential Statement
Theorem: There is no solution in integers to the equation
x2 – y2 =
However, there is a solution in integers to the equation
x2 – y2 =

11. Example: Proving an Existential Statement
The standard 8 x 8 checkerboard, with two adjacent corners removed, can be covered with 1 x 2 and 2 x 1 rectangles.

12. Example: Proving an Existential Statement
The standard 8 x 8 checkerboard, with two adjacent corners removed, can be covered with 1 x 2 and 2 x 1 rectangles.
Yes!

13. Example: Disproving an Existential Statement
The standard 8 x 8 checkerboard, with two opposite corners removed, cannot be covered with 1 x 2 and 2 x 1 rectangles.

14. Example: Disproving an Existential Statement
The standard 8 x 8 checkerboard, with two opposite corners removed, cannot be covered with 1 x 2 and 2 x 1 rectangles.
Oops!

15. Prove or Disprove
Prove or disprove: The standard 8 x 8 checkerboard, with any two squares of different colors removed, can be covered with 1 x 2 and 2 x 1 rectangles.

16. Prove or Disprove
Prove or disprove: For every positive integer n, a 2n x 2n checkerboard, with any two squares of different colors removed, can be covered with 1 x 2 and 2 x 1 rectangles.