Post on 30-Oct-2021
Induction & recursionCS 330 :Discrete structures
Many conjectures have the form : An EMP
Mathematical induction is a technique for proving conjecturesof this form based on the rule of inference :
assuming
fPG) ( basis ] a
amber tknh.fi#gajlmaiaimaep3&IIka7eismfFtwsis.s,prone Plntl) must
arbitrary Pcd , it Pln) be true
depending ondomain of proof
Whit
.info?eyf7anuthahw-?e.canreaehanyrunga and
"
II¥E:q; %5a.tn?:mgIE:*( I
e.
g. provethat Hn (20+2't . . .tt =3"- I)
Pln) : 272't . .- tz"-_zhttl
basis : Plo) : 20=2" '- I
surprisingly!isn't""
uh
whatmene
tryingto
1 = 2 - I = I ✓prove?
(''a'radar
masonryinductive step : show that Pcu)→ Platt)--1272't . .
.tt = It'- I [ inductive hypothesis]
add It ' to both sides
20+2't . . -+It2*1=2*11+2^+1 "
quod wat
=2(It'
) - I denronstrandnm"
=L"- I aE¥orI )
e.
g. provethat Fn Z 5
,2" > n'
Pcn) : 2"7h2
basis : PCs) : 25 > 5232 > 25 ✓
inductive sleep : show Pln)→ pcut 1)
assume Pln) : I > n' [inductive hypothesis]
2*1 = 2 . I
> zu- ( by/]
= ft U2
Z nZ t5h [ since UZ 5 ]> X t2h t I = (ut 1)
2
QED .
e.g.Fibonacci sequence
: F,= I
,E = I
,Fn = Fn - it Fn-z for n >2
I , I , 2 , 3 ,5,8,13
,21,
.--
phone that F, tfz tFst
. . .
t Fac- I= Fak for nZ l
basis : Pci) : F, = I = E V
inductive step : assume Fi tFs t . .. Fac - I
= Fzk
add Fact , to each side
Fi tFs t. . .tfk- I tFzkt I
=Fzkt Fact I
= Fzkt2
=
Fzfkti) QED
e.g. , what is the maximum# of pieces into which we candivide a circularpizza usingen straight arts ?
cats additional total
(n) pieces pieces①
3 311
4 4
⑤5
5 16
③ pieces(n)= It ltzt . .
.tn
= I t(n⇒n= It n-thlinen can intersect z
③ atmost n- I other lines , = n2tnt2④
creating n new piecesI
e-g., pronethat the maximum# of pieces into which we can divide
n't n t 2a circular pizza using n straight
arts is-Z
basis : O cuts = 0tOt2 = I piece VZ
inductive step : assume for u cuts = n2tnzt2 piecesfor ntl outs =n22 t nt l
Z
= n't 3h t4 =n't2ht l tht 3--Z Z
=(nH)Zt (htt)tz-
Z
QED .
e-g. , phonethat all cats are the same size (aka cats are liquid)
Pla) : for any sort of u cats ,all the catsarethe samesin
.
basis : PH V
Tinductive sleep : assume Pcn)
② , Cz , . . . ,Cn are the same size
this worksof ④ ,Cs,. .
.
,cut, are the same sin
PG) → PG) ,--
PG)→P(4) , ..
.
n cats
butwhat aboutwe kno Ice) QE)
pay →plz) ? - takecare tofish outall requiredbase cases !
strong induction is a variant of mathematical widvduinwhere we prone conjectures of the same form (fnE IN PCn) ),but up a different hypothesis in the widedine sleep :
PG) (basis ]
(tKE n P(K)) → Plnti) [inductive dep ]-
tn Pln) ( proof goal ]
Intuition : to provethat we can reach
any rungonand
infinitely tall ladder . .
,-a:*:
:c; ÷:÷:÷s÷;
e.g . every integer n > I is either prime or a product of primes .Pln) = n is eitherprime or a product of primesbasis : PG) - z is prime V
inductive shop : assume Pfk) is true for zs k sn ( inductive
show Rnti) must holdhypothesis]
case l : ntl is prime
case2 : UH is not prime- this means there are two integers a, bwhere n H = ab and 2 E a
,b En
- a,b are prime or product of primes (dueto I.H
.
) so nH is prime or productofprimes .
QED
e.g. , prove that anyamount of n28 can be made with
denominations of 3 and 5
6,°s6, EE
f = 3+5 9 = 3+3+3 10=5+5 11 = 3+3 t5 12 = 3+3+3+3
basis : pls) V
inductive sleep : show that Pcn)→Kuti)
(weak assume Pln) ; to satisfy Parti)induction) case 1 : amount n uses at least one
5
- remove 5 , replace M two 3s
case 2: amount n uses no5s QED .
- remove three 3s (must exist , sincen Z 8) replace M two 5s
e.g. , prove that anyamount of n28 can be made with
denominations of 3 and 5
6,°s6, EE EE
f = 3+5 9 = 3+3+3 10=575 11=3+3 t5 12=3+3+3+3
i.it .basis : PCs)
,Pla)
,Pho)
,PUD , Paz) f
inductive sleep : assume Pcu) istrue for 8 En Ek , where k> 12
(strong induction) to make the amount ktl,we can simply
add 5 to K-4,where PCK-4) is true
due to the t.tt .
QED .
the approach used in a proof by mathematical induction canalso be used to define functions with domain IN .
e.g . flo) = I [ base can] } factorialfunctionf(nti) = (nti) . f(u) ( recursive definition] fun ,
n!
such recursively defined functions are well -defined forall values of the domain .
- induction is a great fit for proving properties offunctions defined in this way !
list the first six terms in the range off :iN→N , where
flo) -- O
f-(NH) -- 2. flu) th
f-6) = o f- (3) = 2 . I t 2=4
fll) = 2 - o to = O f(4) = 2.4 +3 = II
fly = 2 - o t I = I f- (5) = z - Ift 4=26
prone that this function evaluates to I - n - l for all n E IN
flo) = 0
f-(nti) -- 2. flu) t n
basis : flo) = 20- o - I = 0
inductive sleep : flu) = 2"- n - I [ inductive hypothesis]
flat D= 2. (2" - n - 1) tu= zntl - zu - z tu= zn
H- u -Z
= It'- (n+ i) - I QED .