1. 2.3 Methods of Proof

2. 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.

3. 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

4. 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.

5. 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

6. 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.

7. 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)

8. 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

9. 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.

10. 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 = ≠ 3