1. Direct Proof and Counterexample I Lecture 11 Section 3.1 Fri, Jan 28, 2005

2. Definitions A definition gives meaning to a term. A non-primitive term is defined using previously defined terms. A primitive term is undefined. Example A function f : R  R is increasing if f(x)  f(y) whenever x  y. Previously defined terms: function, real numbers, greater than.

3. Definitions Definitions are not theorems. Definitions are often stated in an “if-then” form. Definitions are automatically “if and only if,” even when they aren’t stated that way.

4. Example Definition: A number n is a perfect square if n = k 2 for some integer k. Now suppose t is a perfect square. Then t = k 2 for some integer k. Is this the “error of the converse”?

5. Proofs A proof is an argument leading from a hypothesis to a conclusion in which each step is so simple that its validity is beyond doubt. Simplicity is a subjective judgment – what is simple to one person may not be so simple to another.

6. Types of Proofs Proving universal statements Proving something is true in every instance Proving existential statements Proving something is true in at least one instance

7. Types of Proofs Disproving universal statements Proving something is false in at least one instance Disproving existential statements Proving something is false in every instance

8. 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 in D (generic particular). Assume that P(x) is true (hypothesis). Argue that Q(x) is true (conclusion).

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

10. Example: Direct Proof If 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. Then 2k 2 + 1 = 2(3m  1) 2 + 1 = 18m 2  12m +3 = 3(6m 2  4m + 1). Therefore, n 3 – n is a multiple of 12.

11. An Alternate Proof Proof: n 3 – 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, n 3 – n must be a multiple of 12.

12. Example: Direct Proof Theorem: If x, y  R, then x 2 + y 2  2xy. Incorrect proof: Let x, y  R. x 2 + y 2  2xy. x 2 – 2xy + y 2  0. (x – y) 2  0, which is known to be true. What is wrong?

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