All About Proofs

Lecture 2: Sep 5 (based on slides in MIT 6.042)

2 2 2a b c

Familiar?

Obvious?

cb

a

Yes!

No!

Pythagorean theorem

cb

a

Rearrange into: (i) a cc square, and then (ii) an aa & a bb square

A Cool Proof

c

cc

a b

c

b-a

A Cool Proof

cb

a b-a

b-a

A Cool Proof

ba

a

ab-a

A Cool Proof

74 proofs in http://www.cut-the-knot.org/pythagoras/index.shtml

b

1110

11

11

11

11

11

10

Getting Rich By Diagram

11

Profit!

10

10

11

11

11

11

Getting Rich By Diagram

Getting Rich By Diagram

The bug:

are not right triangles!

The top and bottom line of the “rectangle” is not

straight!10

11 1

1

1

Getting Poor By Diagram

Another Exercise

Evidence vs. Proof

Let p(n) ::= n2 + n + 41.

Claim: p(n) is a prime number for all nonnegative integer n.

Check: p(0) = 41

p(1) = 43

p(2) = 47

p(3) = 53

p(20) = 461

p(39) = 1601

prime

prime

prime

prime

prime

prime

looking good!

enough already!

Evidence vs. Proof

Let p(n) ::= n2 + n + 41.

Claim: p(n) is a prime number for all nonnegative integer n.

Well, it is clear that the claim is true

But NO! p(40) = 1681 is not a prime number. Why?

p(40) = 402 + 40 + 41

= 402 + 2x40 + 1

= (40+1)2

Evidence vs. Proof

Let p(n) ::= n2 + n + 41.

Actually, p(n-1) = p(-n), so if we consider

f(n) = p(n - 40) = n2 + 79n + 1601

f(n) is a prime for x=0,1,2,…,79

This is the current champion (the function n2 - 2999n + 224851

also yields 80 consecutive primes from 1460 to 1539).

More Claims

Claim: is a prime number for all nonnegative n.

Fermat number f(0)=3, f(1)=5, f(2)=17, f(3)=257, f(4)=65537,

f(5)=4294967297

The first 49 Fermat numbers haven been checked,

but except for those up to f(4) every one is composite (not prime)!

Euler conjecture:

has no solution for a,b,c,d positive integers.

Open for 218 years,until Noam Elkies found

4 4 4 495800 217519 414560 422481

Even More Claims

Fermat (1637): If an integer n is greater than 2,

then the equation an + bn = cn has no solutions in non-zero integers a,

b, and c.Andrew Wiles (1994): prove it using “elliptic curves”.

http://en.wikipedia.org/wiki/Fermat's_last_theorem

Claim: has no solutions in non-zero integers a, b, and c.

False. But smallest counterexample has more than 1000 digits.

Goldbach’s conjecture: Every even number is the sum of two prime numbers.

Proving an Implication

Goal: If P, then Q. (P implies Q)

Method 1: Write assume P, then show that Q logically follows.

IfClaim: , then

Reasoning: When x=0, it is true.

When x grows, 4x grows faster than x3 in that range.

Proof:

When

(see page 19 of the book)

Proving an Implication

Claim: If r is irrational, then √r is irrational.

How to begin with?

What if I prove “If √r is rational, then r is rational”, is it equivalent?

Yes, this is equivalent;

proving “if P, then Q” is equivalent to proving “if not Q, then not P”.

Goal: If P, then Q. (P implies Q)

Method 1: Write assume P, then show that Q logically follows.

Proving an Implication

Claim: If r is irrational, then √r is irrational.

Method 2: Prove the contrapositive, i.e. prove “not Q implies not P”.

Proof: We shall prove the contrapositive –

if √r is rational, then r is rational.

Since √r is rational, √r = a/b for some integers a,b.

So r = a2/b2. Since a,b are integers, a2,b2 are integers.

Therefore, r is rational.

(Q.E.D.) "which was to be demonstrated", or “quite easily done”.

Goal: If P, then Q. (P implies Q)

Proving an “if and only if”

Goal: Prove that two statements P and Q are “logically equivalent”,

that is, one holds if and only if the other holds.

Example:

An integer is a multiple of 3 if and only if the sum of its digits is a multiple of 3.

Method 1: Prove P implies Q and Q implies P.

Method 1’: Prove P implies Q and not P implies not Q.

Method 2: Construct a chain of if and only if statement.

(see page 21 of the book)

Propositional (Boolean) Logic

Proposition is either True or False

Examples:

Non-examples:

Hello.

How are you?

True

False

2 + 2 = 4

3 x 3 = 8

787009911 is a prime

Logic Operators

Connecting propositions.

F

F

F

T

P Q

FF

TF

FT

TT

QP

NOT::

AND::

F

T

T

T

P Q

FF

TF

FT

TT

QP

OR::

coffee “or” tea

exclusive-or

IMPLIES:: IFF::

Logic Operators

T

T

F

T

P Q

FF

TF

FT

TT

QP

Convention: if we don’t say anything wrong, then it is true.

T

F

F

T

P Q

FF

TF

FT

TT

QP

Note: P Q is equivalent to (P Q) (Q P)

Note: P Q is equivalent to (P Q) ( P Q)

Math vs English

Parent: if you don’t clean your room, then you can’t watch a DVD.

C D

This sentence says

In real life it also meansSo

Mathematician: if a function is not continuous, then it is not differentiable.

This sentence says

But of course it doesn’t mean

Logical Deduction

From: P implies Q, Q implies R

To: P implies R

conclusion

antecedents

Definition: A rule is sound if the conclusion is true

whenever all antecedents are true.

More Examples

sound

sound

sound

sound

unsound

READ CHAPTER 1!!