1. Proof Methods

2. Introduction Formal Proofs vs. Informal Proofs
Many proof methods exist however none is guaranteed to prove a true theorem
i.e., no general proof algorithm exists
Rather we must intelligently choose our proof technique based on the particulars of the problem
Exhaustive, direct, contraposition, contradiction, induction
Important Terms
Inductive Reasoning – a general conclusion is drawn from a set of experiences.
Deductive Reasoning – a specific conclusion is shown to follow from a set of premises
Hypothesis
generation
Hypothesis
confirmation

3. Direct Proof
To prove P  Q, assume P is true and use formal logic to show that Q is also true
Example: Our previous proofs were direct proofs
Problem: Too slow and cumbersome to be practical
Can use same idea of direct proof but less formal notation (called informal direct proof)
Example: Conjecture: “If an integer is divisible by 6, then twice that integer is divisible by 4” (taken from text, p.93)
Informal Proof

4. Counterexample
A single counterexample to a conjecture is sufficient to disprove that conjecture
Example: Finding large prime numbers (for cryptography)
We observe that f(n) = n2 – n + 41 produces prime numbers for n = 1 (41), 2 (43), 3 (47), and 4 (53)
Conjecture: f(n) is prime for all positive integers n.
Counterexample: n = 41
f(41) = 412 – = 412

5. Exhaustive Proof Can one prove a conjecture by example? In general, no
In rare cases, yes
Conjecture deals with finite collection
Can show conjecture true for each element in collection
Called proof by exhaustion
Example: f(n) is prime where 0  n  40

6. More Examples Prove or disprove for all positive integers.
If 2n + 3 is odd then the number n is odd.
If 3n + 2 is odd then the number n is odd.
If x + y = 7, then either x or y must be even.

7. Contraposition Proof To prove P  Q, prove that Q  P Converse
(Q  P)  (P  Q) is a tautology
Example: n2 is odd  n is odd
Contrapositive: n is even  n2 is even
Proof:
Converse
The contrapositive is not the converse
Converse of Q  P is P  Q
“If and only if” conjectures require
Proof of the implication
Proof of the converse
Examples
If x is positive then so is x + 1.
If the product of 2 integers is not divisible by n then neither integer is.

8. Contradiction Proof Example: Prove that all numbers are interesting
Not an implication: Assume Q
What do we mean by interesting?

9. Contradiction Proof
A contradiction is a wff whose truth value is always false
Example: (A  A)
Notice the following are tautologies
(Q  false)  (Q)
(P  Q  false)  (P  Q)
Try to develop the truth table for the above formulae
To prove a conjecture true, prove that the negation of the conjecture is false
To prove a simple Q, assume Q and show that this leads to a contradiction.
To prove an implication P  Q, assume P and Q and show that this leads to a contradiction
Remember this doesn’t prove that Q is true
only that P  Q is true

10. Contradiction Proof Example: Prove that if 3n + 2 is odd then n is odd
Let P be “3n + 2 is odd”
Let Q be “n is odd”
Assume P  Q
More Examples
No integer is both even and odd.
If the sum of two primes is prime, then one of the primes must be 2.

11. Practice Problems Mathematical Structures for Computer Science
Section 2.1: 12, 14, 15, 19, 24, 26, 35, 36, 38, 40, 47, 48, 51, 55, 62
Section 2.2: 1, 5, 8, 12, 13, 16, 22, 29, 36, 43, 44, 51, 66, 74

12. Project 1

13. Contradiction Proof Example: The Halting Problem Program
“There is an algorithm that takes as input a computer program and input to that program and determines whether the program will eventually stop when run with this input.”
Recall that an algorithm must terminate, therefore you can’t just run the program on the input to see if it terminates – what if it doesn’t?
Program
Halting Algorithm
Halt/Never
Input

14. The Halting Problem P H(P, I) Halt/Never I
Assume there is a solution to the Halting Problem,
call it H(P, I)
H(P,I) == “halt” when P terminates on I
H(P,I) == “never” when P does not terminate on I
We need a way to represent the Program and Input when it’s given to the Halting Algorithm.
Represent both using 0s and 1s
So we can run H(P,P) and according to our assumption, it will return either “halt” or “never”
H(P, I) P I
Halt/Never

15. The Halting Problem Consider the following algorithm K(P)
temp = H(P,P)
if temp == “halt” then
while (true) do {}
end if
What happens if we try K(K)?
if H(K,K) is “never”, then by definition K(K) halts
a contradiction
if H(K,K) is “halt”, then by definition K(K) doesn’t halt
Therefore, no such Halting Algorithm can exist!
Often used in computer science as the base line
“If we could write program X we could write H and therefore program X is not possible.”
H(K, K) K
Halt/Never

16. More Proof Methods
If the proof of a theorem is not immediately apparent, it may be because you are trying the wrong approach….
Proof by calculus: "This proof requires calculus, so we'll skip it." Proof by lack of interest: "Does anyone really want to see this?" Proof by illegibility: Proof by logic: "If it is on the problem sheet, it must be true!" Proof by clever variable choice: "Let A be the number such that this proof works..." Proof by tessellation: "This proof is the same as the last." Proof by divine word: "...And the Lord said, 'Let it be true,' and it was true." Proof by stubbornness: "I don't care what you say- it is true." Proof by simplification: "This proof reduced to the statement = 2." Proof by hasty generalization: "Well, it works for 17, so it works for all reals." Proof by deception: "Now everyone turn their backs..." Proof by supplication: "Oh please, let it be true." Proof by poor analogy: "Well, it's just like..." Proof by avoidance: Limit of proof by postponement as it approaches infinity Proof by design: If it's not true in today's math, invent a new system in which it is. Proof by authority: "Well, Don Knuth says it's true, so it must be!" Proof by intuition: "I have this gut feeling."
Proof by obviousness: "The proof is so clear that it need not be mentioned." Proof by general agreement: "All in favor?..." Proof by majority rule: Only to be used if general agreement is impossible. Proof by imagination: "Well, we'll pretend it's true..." Proof by convenience: "It would be very nice if it were true, so..." Proof by necessity: "It had better be true, or the entire structure of mathematics would crumble to the ground." Proof by plausibility: "It sounds good, so it must be true." Proof by intimidation: "Don't be stupid; of course it's true!“ Proof by terror: When intimidation fails... Proof by lack of sufficient time: "Because of the time constraint, I'll leave the proof to you." Proof by postponement: "The proof for this is long and arduous, so it is given to you in the appendix." Proof by accident: "Hey, what have we here?!" Proof by insignificance: "Who really cares anyway?“ Proof by profanity: (example omitted) Proof by definition: "We define it to be true." Proof by tautology: "It's true because it's true." Proof by plagiarism: "As we see on page 289,..." Proof by lost reference: "I know I saw it somewhere...."

17. Induction Proof Highly useful proof method in computer science
used in statement counting
used as basis of recursion
Induction involves the properties of positive integers
Proof by induction involves two steps
Prove conjecture for the first case
Prove that if the conjecture is true for a given case, it is true for the next case
Leads to the following first principle of induction
P(1) is true
k P(k) true  P(k+1) true
Basis step
Inductive step
P(n) true for all positive integers n.

18. Induction Proof
Example: Use Induction to prove that n < 2n for all positive integers n.
Basis: P(1): 1 < 2
Induction: P(k+1): Assume that if k < 2k, we can prove (k+1) < 2(k+1)
Note: Don’t be confused – mathematical induction is a
deductive technique, not a method of inductive reasoning!

19. Induction Proof Example: Use Induction to prove that
for all positive integers n.
Basis: 1 = 1(2) / 2
Induction:
Basic Process:
1. Establish theorem
2. Show true for base case
3. Assume inductive hypothesis LHSih = RHSih
4. State “want to show” as LHSwts = RHSwts
5. Give definition of LHSwts
6. Translate to use LHSih
7. Substitute RHSih for LHSih
8. Manipulate to find RHSwts

20. Induction Proof More Examples ∑ 5i= 5n (n+1)/2 1 ≤ i ≤ n
(b) ∑ i3 = [n2 (n+1)2]/ ≤ i ≤ n

21. Induction Proof
Sometimes it is convenient to use the second principle of induction
P(1) is true
k[ P(r) true for 1  r  k  P(k+1) is true ]
Example: Show that if n is an integer greater than 1, then n can be written as the product of primes.
Let P(n) be the statement that “n can be written as a product of primes”
Basis: P(2) = 2
Induction: Assume P(i) is true for all integers 1 < i  k.

22. Induction Proof
Example: Your bank ATM delivers cash using only $20 and $50 bills. Prove that you can collect, in addition to $20, any multiple of $10 that is $40 or greater (text, p. 117).

23. Practice Problems Mathematical Structures for Computer Science
Section 2.1: 12, 14, 15, 19, 24, 26, 35, 36, 38, 40, 47, 48, 51, 55, 62
Section 2.2: 1, 5, 8, 12, 13, 16, 22, 29, 36, 43, 44, 51, 66, 74