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Methods of Proof

Lecture 3: Sep 9

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This Lecture

Now we have learnt the basics in logic.We are going to apply the logical rules in proving athe atical theore s.

! "irect proof

! #ontrapositive

! Proof by contradiction

! Proof by cases

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Basic Definitions

$n integer n is an even nu ber

if there e%ists an integer & such that n ' (&.

$n integer n is an odd nu ber

if there e%ists an integer & such that n ' (&)*.

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Proving an Implication

+oal: ,f P- then . /P i plies 0

Method *: Write assu e P- then show that logically follows.

1he su of two even nu bers is even.

% ' ( - y ' (n

%)y ' ( )(n

' (/ )n0

Proof

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Direct Proofs

1he product of two odd nu bers is odd.% ' ( )*- y ' (n)*

%y ' /( )*0/(n)*0

' 2 n ) ( ) (n ) *

' (/( n) )n0 ) *.

Proof

,f and n are perfect s uare- then )n)(4/ n0 is a perfect s uare.

Proof ' a ( and n ' b ( for so e integers a and b

1hen ) n ) (4/ n0 ' a ( ) b ( ) (ab

' /a ) b0 (

So ) n ) (4/ n0 is a perfect s uare.

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This Lecture

! "irect proof

! #ontrapositive


! Proof by cases

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Proving an Implication

#lai : ,f r is irrational- then 4r is irrational.

5ow to begin with6

What if , prove 7,f 4r is rational- then r is rational8- is it e uivalent6

es- this is e uivalent- because it is the contrapositive of the state ent-

so proving 7if P- then 8 is e uivalent to proving 7if not - then not P8.


Method *: Write assu e P- then show that logically follows.

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Rational Number

is rational there are integers a and b such that

and b ;

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Proving the Contrapositive

#lai : ,f r is irrational- then 4r is irrational.

Method (: Prove the contrapositive - i.e. prove 7not i plies not P8.

Proof: We shall prove the contrapositive B 7if 4r is rational- then r is rational .8

Since 4r is rational- 4r ' a@b for so e integers a-b.

So r ' a ( @b( . Since a-b are integers- a ( -b( are integers.

1herefore- r is rational.

/ .C.".0 Dwhich was to be de onstratedD- or 7 uite easily done8.


.C.".

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Proving an “if and only if

+oal: Prove that two state ents P and are 7 logically e!uivalent - that is- one holds if and only if the other holds.

C%a ple: Eor an integer n- n is even if and only if n ( is even.

Method *a: Prove P i plies and i plies P.

Method *b: Prove P i plies and not P i plies not .

Method (: #onstruct a chain of if and only if state ent.

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Proof the Contrapositive

State ent: ,f n ( is even- then n is even

State ent: ,f n is even- then n ( is even

n ' (&

n( ' 2& (

Proof:

Proof: n( ' (&

n ' 4/(&0

66

Eor an integer n- n is even if and only if n ( is even.

Method *a: Prove P i plies and i plies P.

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Since n is an odd nu ber- n ' (&)* for so e integer &.

So n( is an odd nu ber.

Proof the Contrapositive

State ent: ,f n ( is even- then n is even

#ontrapositive: ,f n is odd- then n ( is odd.

So n ( ' /(&)*0 (

' /(&0 ( ) (/(&0 ) *

Proof /the contrapositive0:

Method *b: Prove P i plies and not P i plies not .

Eor an integer n- n is even if and only if n ( is even.

' (/(& ( ) (&0 ) *

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This Lecture

! "irect proof

! #ontrapositive


! Proof by cases

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F P

P

→

Proof by Contradiction

1o prove P- you prove that not P would lead to ridiculous result-

and so P ust be true.

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! Suppose was rational.

! #hoose - n integers without co on pri e factors /always possible0

such that

! Show that and n are both even- thus having a co on factor (-

a contradiction F

n

m=2

Theorem" is irrational .2

Proof /by contradiction0:


2

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l m 2=so can assu e2 24m l =

22 2 l n =

so n is even .

n

m=2

mn =2

22

2 mn =

so is even .

2 22 4n l =


Theorem" is irrational .2

Proof /by contradiction0: Want to prove both and n are even.

ecall that is even if and only if (

is even.

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Infinitude of the Primes

Theorem# 1here are infinitely any pri e nu bers.

$ssu e there are only finitely any pri es.

Let p *- p( - A- pN be all the pri es.

/*0 We will construct a nu ber N so that N is not divisible by any p i.

Gy our assu ption- it eans that N is not divisible by any pri e nu ber.

/(0 Hn the other hand- we show that any nu ber ust be divided by so e pri e.

,t leads to a contradiction- and therefore the assu ption ust be false.

So there ust be infinitely any pri es.


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Divisibility by a Prime

Theorem# $ny integer n I * is divisible by a pri e nu ber.

,dea of induction.

! Let n be an integer.

! ,f n is a pri e nu ber- then we are done.

! Htherwise- n ' ab- both are s aller than n.! ,f a or b is a pri e nu ber- then we are done.

! Htherwise- a ' cd- both are s aller than a.

! ,f c or d is a pri e nu ber- then we are done.

! Htherwise- repeat this argu ent- since the nu bers are getting s aller and s aller- this will eventually stop and we have found a pri e factor of n.

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Infinitude of the Primes

Theorem# 1here are infinitely any pri e nu bers.

Claim: if p divides a- then p does not divide a)*.

Let p *- p( - A- pN be all the pri es.

#onsider p *p( ApN ) *.



a ' cp for so e integer c

a)* ' dp for so e integer d

'I * ' /d>c0p- contradiction because pI'(.

So- by the clai - none of p *- p( - A- pN can divide p *p( ApN ) *- a contradiction.

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This Lecture

! "irect proof

! #ontrapositive


! Proof by cases

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Proof by Cases

% is positive or % is negative

e.g. want to prove a non?ero nu ber always has a positive s uare.

if % is positive- then % ( I

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The $!uare of an %dd Integer

3 ( ' 9 ' =)*- J ( ' (J ' 3%=)* AA *3* ( ' *K* * ' (*2J%= ) *- AAA

,dea *: prove that n ( B * is divisible by =.

,dea (: consider /(&)*0(

,dea

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Rational vs Irrational

uestion: ,f a and b are irrational- can ab be rational66

We /only0 &now that 4( is irrational- what about 4( 4( 6

Case &" '( '( is rational

1hen we are done- a'4(- b'4(.

Case (" '( '( is irrational

1hen / '( '( )'( ' '( ( ' (- a rational nu berSo a' '( '( - b' 4( will do.

So in either case there are a-b irrational and a b be rational.

We don t /need to0 &now which case istrueF

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$ummary

We have learnt different techni ues to prove athe atical state ents.

! "irect proof

! #ontrapositive


! Proof by cases

Ne%t ti e we will focus on a very i portant techni ue- proof by induction.

l03 dis.math

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