[0:00]Hello, everybody. Uh before we get started, just a quick announcement about office hours, if you have any questions or want to go over anything or proofs, Uh my office hours are going to be held in this building room 218 for office hours. A statement is a sentence that can be declared either true or false. A proof of a statement is a series of steps establishing beyond doubt that the statement is true. Type of statement: If P then Q. How to prove: Suppose P is true, use this to prove Q. "Contrapositive": suppose Q is false, use this to prove P is false. P implies Q. P => Q Example: Cancellation Law. Let F be a field. Let a,b,c be in F, with a != 0. If ba = ca then b = c. Proof: Let a be in F. Suppose that ba = ca. Let a' be the multiplicative inverse of a.
[15:00]Then (ba)a' = (ca)a'. b(aa') = c(aa') by associativity. b*1 = c*1 since a,a' are mult. inverses. b = c since 1 is the mult. identity.
[20:07]P and Q. Show P is true. Show Q is true. P or Q. Assume P is false, use this to prove Q. Exercise: prove sqrt(2) is not equal to Q. Assume sqrt(2) = a/b.
