2.3 Methods of Proof

Methods of Proof -Direct method -Indirect Methods: Showing a contrapositive Proof by contradiction Proof by counter example A proof must demonstrate that a statement is true for all cases (a tautology) We follow rules of inference for proofs. Rules of inference include a premise and a conclusion.

A mathematical proof: -must begin with a hypothesis or premises -proceed thru various steps, justified by some rules of inference -and arrive at a conclusion

A direct method of proof uses a previously proven fact. If p ⇒ q (if p then q) is a tautology, we say that q logically follows p. If (p1 ∧ p2 ∧… .∧ p n ) ⇒q is a tautology We can write: P1 P2 P3 . This section is the hypothesis or premises . P n ______________ This line means if…then  q Therefore q ( means therefore) q is the conclusion This shows that if Px is true, then q is true.

Page 60 (( p ⇒ q) ∧ (q ⇒ r)) ⇒ (p ⇒ r) is a tautology IF ((IF p then q) and (IF q then r)) then (IF p then r) We can rewrite: p ⇒ q If p then q q ⇒ r If q then r ________ if ….then  p ⇒ r Therefore If p then r

Following is an argument for this tautology: If you invest in the stock market, then you will get rich If you get rich, then you will be happy _______________________________________________  If you invest in the stock market, then you will be happy The argument is valid however, the conclusion may be false.

Indirect method showing contrapositive Following is a tautology representing contrapositive (p ⇒ q) ⇔ ((˜ q ⇒ (˜ p)) (IF p then q) IF and only IF (if not q then not p) n is an integer Prove that if n2 is odd then, n is odd p: n2 is odd q: n is odd We need to prove p ⇒ q is true…. instead, we prove the contrapositive: IF n is even, n2 is even. n = 2K where K is an integer. We use 2 because 2 is a factor for all even numbers. n . n = (2k) . (2k) n2= 2 (2.k.k) n2 = 2 (some integer)

Indirect method proof by contradiction This method is based on the tautology ((p ⇒ q) ∧ (˜ q)) ⇒ (˜ p) If (If p then q and not q) then not p We can write: p ⇒ q If p then q ˜ q not q _______ IF….then  ˜ p Therefore not p

To prove by contradiction, we went to prove that something is not true and show that the consequences are not possible. The consequences contradict what we assume. If I get a parking ticket and I didn’t pay it, I would get a nasty letter from the city. We know that I didn’t’ get a nasty letter from the city. We can deduce that I paid my ticket since I didn’t get a nasty letter from the city. To prove by contradiction, we would ASSUME that I did not pay the ticket and deduce that I should get a nasty letter. However, we know that I didn’t get a nasty letter. This statement is a contradiction and therefore our assumption is wrong.

Indirect method proof by counter example Prove or disprove If x and y are real numbers, (x2 = y2) ⇔ (x=y) ⇔ means IF and only IF We could find many examples to support this statement We only need one example to disprove this statement Counter Example: (-3)2 = 32 -3 ≠ 3