Direct Proof and Counterexample I

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

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.

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.

Example • Definition: A number n is a perfect square if n = k2 for some integer k. • Now suppose t is a perfect square. • Then t = k2 for some integer k. • Is this the “error of the converse”?

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.

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

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

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 arbitraryx in 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 • 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 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. • Incorrect proof: • Let x, yR. • x2 + y2 2xy. • x2 – 2xy + y2 0. • (x – y)2 0, which is known to be true. • What is wrong?

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

Direct Proof and Counterexample I