Direct Proof and Counterexample I Lecture 12 Section 3.1 Tue, Feb 6, 2007

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 Statements to Prove Proving universal statements Proving something is true in every instance Proving existential statements Proving something is true in at least one instance

Types of Statements to Prove Disproving universal statements Proving something is false in at least one instance Disproving existential statements Proving something is false in every instance

Example Theorem: For every function f(x) = mx + b and for every function g(x) = rx 2 + sx + t, there are constants c and n such that for all x > n, f(x) < cg(x).

Example Theorem: For every function f(x) = mx + b and for every function g(x) = rx 2 + sx + t, there are constants c and n such that for all x > n, f(x) < cg(x).

Example Theorem: If S is an infinite regular set of strings, then there is an integer n > 0 such that for all strings w in S with |w| > n, there exist strings x, y, z such that w = xyz and for all integers i  0, xy i z is in S.

Example Theorem: If S is an infinite regular set of strings, then there is an integer n > 0 such that for all strings w in S with |w| > n, there exist strings x, y, z such that w = xyz and for all integers i  0, xy i z is in S.

Example Theorem: If S is an infinite regular set of strings, then there is an integer n > 0 such that for all strings w in S with |w| > n, there exist strings x, y, z such that w = xyz and for all integers i  0, xy i z is in S.

Proving Existential Statements Proofs of existential statements are often called existence proofs. Two types of existence proofs Constructive Find an instance where the statement is true. Non-constructive Argue indirectly that the there must be an instance where the statement is true.

Example: Constructive Proof Theorem: Given a segment AB, there is a midpoint M of AB. AB

Example: Constructive Proof Theorem: Given a segment AB, there is a midpoint M of AB. Proof: Draw circle A. AB

Example: Constructive Proof Theorem: Given a segment AB, there is a midpoint M of AB. Proof: Draw circle A. Draw circle B. AB

Example: Constructive Proof Theorem: Given a segment AB, there is a midpoint M of AB. Proof: Draw circle A. Draw circle B. Form  ABC. AB C

Example: Constructive Proof Theorem: Given a segment AB, there is a midpoint M of AB. Proof: Draw circle A. Draw circle B. Form  ABC. Bisect  ACB, producing M. AB M C

Justification Argue by SAS that triangles ACM and BCM are congruent and that AM = MB. AB M C

Example: Constructive Proof Theorem: The equation x 2 – 7y 2 = 1. has a solution in positive integers.

Example: Non-Constructive Proof Theorem: There equation x 5 – 3x + 1 = 0 has a solution.

Disproving Universal Statements The negation of a universal statement is an existential statement. Construct an instance for which the statement is false. Also called proof by counterexample.

Example: Proof by Counterexample Theorem: The equation is not true in general.

Example: Proof by Counterexample Theorem: The inequality is not true in general.