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

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).

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.

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.

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.

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.

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?”

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

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

Example: Disproving an Existential Statement Theorem: There is no solution in integers to the equation x2 – y2 = 101010 + 2. However, there is a solution in integers to the equation x2 – y2 = 101010 + 1.

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.

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!

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.

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!

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.

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.