Monday .

I . Go through syllabusL

.

--. . .

-

Div & modsa

Nd = theintegerq"uotienQnm = the integer remaindern = d.qtr . as red

+ Tquotient remainder

Example :

1. Compute 32 div 9 .=3

32 mod 9 ,

= 5C

32 e 9 .

-

T-

a-27 2×9 485

3×9 = 274*9 = 36

2.MtZ

. if Mmoidbwhat is Eh mod① ?!EEE '

M=HE

⇒ 4M ;÷÷÷:÷¥ "km

Ex : tf n . dt2 , d to

d l h ⇐ n mod d = O-

H"

n is multiple of d"

H:n is odd ⇒ n mod 2 - lH is even ⇐ /nmad2ITf-

Thin.The parity of Consecration Integers.tweeters has

opposite parity .

root:*.:÷*÷÷÷÷÷Mtl = 28+2Chimed z IiftoMtl is even

2 . If M is even

similar - - - -

M mad 2=0!

Mtl Mod 2 = I ⇒ -MH is odd.

a. E. is.

-• Method of proof by division into Cases

.

I.""'÷¥¥¥tAsEEve the following :

A

". If A.

,then C )2! If th

,

s. . :* .

÷: :

Wednesday :- Method of proof by division into cases

Ex.( Representation of Integers )REZ

,

÷ : :: : ti:::-

Ex. ( Rewrite Integers Modulo 4).

kt④÷÷÷÷÷÷÷÷¥÷÷:÷÷÷÷i÷÷

Ex. Th m

.The square of myotonia

has the form for some MEZ.

proof I Assume k is odd .

by def k= for some NEZ '

R' = (2ND'

- 4nh4W8-M

proof by division cases :

1. be 4Mt I AZ= mk! (4M¥- 16mF Imei 8km:mln ✓

2. KIM 's

fist '

k¥14men's Ibm't 24mi- 9=ItnE'D I

a.ED.

• Def : txt IR,the absolute value of X ,

denoted as 1×1,

is defined as

1×1=4 X i ifxo

- X, if xao .

-4;÷÷ .

we " -¥¥⑤1 : rye ,

Ink r.

s: :i?÷÷.- -

- - -

t. .

..

InEIR. I -rt

proofs i. no→g÷÷÷Hmv

Thin.the Triangle laeqenditgl.

FEHER . III 's KKKproof:

'##×¥y l ? xtyzo.se/XtyI=xty

Want to prove Xty Efxltlgl .By lemur 't X Eld

, y e ly I

go,

add the above Inequalities , xty stay,y%.

Done !

Xtyco . So lxtyf. - City)Want to prove - ( xtylstxltlyl Lwhich is equivalent to -X - y④④Hyliii. in.

€00-

- txt

*III.' '

up*yea.my/..

QED.

←

-

/

§4.

lndirectttrgament.TLContradiction And Contraposition .

- proof by Contradiction1. Suppose the statement is false-L.Under such assumption . it leads Contradictions

.

Why ? ftp.T#ntegeIeeisNOogaestSupposeN0T@Then Thereisagneatestuiteger⇒ KEI s.t.V-ltzfhenlt-pf.IQEEE.EE#EkeookDWe find a Contradiction . QED

Thin.

Sum of a national * and

(az irisational tf is irrational .HIE its is irrational=

=

rts is irrational .)

2-

Proof :Su@ppoaNOTa.s

::¥nEi÷¥rational

res C- Qr t Q }⇒ SHED -

e

Q is closed under subtraction SEALwe find a Contradiction "

QED.

Friday : -proafbycontrapositionfWrite statement as

Mentally

yVXED , iffy then In2

.Consider the Contraposition(AXED , if ndaa then npcx)

3- Prove the contraposition-

ForIgn . one ,

o..:c.÷:÷÷÷÷i¥::Suppose NEX .nfor some Rez

.

N' = 6kt it - 4h44k "= 212k¥24 ) -11

t.by def ti is odd QED.

proof:C Bg Contradiction)-

Suppose NOT .F- NEZ , St. Wiseman

bat nisodd.TW""'n:÷i÷÷÷::

⇒ (zkkck) t '

n- is a odd bg del

So We have a Contradiction DIED.

-Remark : l

. proof by Contradiction is More general2 . Use indirect method when ~ .AM

is easy to deal with .

W

-

§4I Indirect Argument :Two Classical Thin

.

Hippasus • Thon. T2 is irrational .

Thetapartum

Pr:t.supposeuot.GEis national. ice .

IIs µ- ,for some Mi NEZ

,

µ

ta.i%::÷..f MZ avi is even.

By thin . mi¥mza for some k

4k¥ 2ndI

Bsrnm"=nYseI

.

"

That.is#n;:YionQED

.

Ex: It 3 if is irrational .

proof.. Suppose NOT .

Then HSE is rational.

"" t!¥÷F for some math.

*m⑤?tf is rational . Because QC is closed

Under subtraction&division

We get a Contradictionto ThurI

. QED..

Question :Are there infinitely many primes ?

Thin. Yes

.

there are infinitely my prime .

Prod.

Fat Z, prime number P.

=

if Plathen p f caus-

proof; Suppose NOT.

Then I AEK . paine # p sit .

Pla and of catsThen Pla ⇒ A- P . k for some KEI , ✓

Play ⇒ atTl for som lez . ✓1-- pal - Pak =p (e - k) .

l- KEK✓

⇒ plot

By Thin , factor of one is 1 or- l

.

p =L or -1

but p is prime . Contradict. !-

Thru.

Yes.

there are infinitely my pain .poof: sjeoaa.tw?auteaitebmasinim@

We can list them from small to large2 .

tgkBetztj.Q@DXIfEEExhtIE2l.Xs1

.

Claim !Suppose NOT.enHETeEiha.⇐

Now ④ is prime .

④ CREW -11

t④ Contradiction !