L03 Dis.math
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Transcript of L03 Dis.math
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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 contradiction
! 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.
+oal: ,f P- then . /P i plies 0
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.
+oal: ,f P- then . /P i plies 0
.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 contradiction
! 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:
Proof by Contradiction
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 =
Proof by Contradiction
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.
Proof /by contradiction0:
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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 ) *.
Proof /by contradiction0:
Proof /by contradiction0:
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 contradiction
! 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 contradiction
! Proof by cases
Ne%t ti e we will focus on a very i portant techni ue- proof by induction.