too ;oioiF. e . . - ej-n.CO/000)- ( o R2 0 ! o l t - oFEtot - = - - PDF document

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too ;oioiF. e . . - ej-n.CO/000)- ( o R2 0 ! o l t - oFEtot - = - - PDF document

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HONORS COMBINATORICS SPRING 2020 Week 1 slides April 79 04-07 class page 1/4 H " R Z . . - xn Xo - xjlkt Heli too ;oioiF. e . . - ej-n.CO/000)- ( o R2 0 ! o l t - oFEtot - = T2 htt hyetp lane in h =D 04-07 class

slide-1
SLIDE 1

HONORS COMBINATORICS

SPRING 2020

Week 1 slides April 7–9
slide-2
SLIDE 2 04-07 class page 1/4

H

R

"

Z

Xo

. . - xn

Heli

  • xjlkt

e.

. -

too;oioiF¥.¥

ej-n.CO/000)-

(o

0! o l

R2 t

  • oFEtot¥
= T2
slide-3
SLIDE 3 04-07 class page 2/4

hyetplane in

htt

h

=D

do

i
  • yea

(O

. .0,10.
  • O)
  • ↳is

I

p,

. - pm E IR

" at pairwise

equal dist ⇒ m Intl

slide-4
SLIDE 4 04-07 class page 3/4

pic.

  • pm EIR

"

GEER

" pi -pjHE{sit}

→

h=

I

11-1

m =-3

TT

÷¥¥E

is this

Max

?

slide-5
SLIDE 5 04-07 class page 4/4

2 -distance set

in

HR

"

tee

.

disease.as

slide-6
SLIDE 6 04-08 PROBLEM SESSION Page 1/5

2

  • distance

SEIR

"

ISI >

en'

1st

  • (I)=Nz
slide-7
SLIDE 7 04-08 problem session page 2/5

f- I g

polynomials

divides

:

Over IR

⇐ e)( g

  • f. h)

prime

  • exponent poly
:

f-(x)

= 3.It 4×19-8×37

PROVE :# f-tokzg-topn.me

  • expflg)
slide-8
SLIDE 8 04-08 problem session Page 3/5

A

iii.

'

i'iii.to

=

%

. ' to
slide-9
SLIDE 9 04-08 problem session page 4/5

±

.

÷i

Eee

%÷÷÷%i÷

.

slide-10
SLIDE 10 04-08 problem session page 5/5 I

2

3

4

1234

④ 3452¥

, { Lk

3

4 56

0000

✓5670

4444

slide-11
SLIDE 11

04-09 CLASS

Page 1/20

A- { a ,b , c}

=

{ a , b. a , c. c}

A

  • B

if

(Hx)CxeA←sxEB)

( Al =3

cardinality

  • f A
slide-12
SLIDE 12 04-09 class page 2/20

f

: # → B

Akaka}

13=112

A

: domain

A

B : codomain a#

b

T z

range (f) ={?fE

, c
  • it

C- codomain

slide-13
SLIDE 13 04-09 class page 3/20

BA -{f

: A → B}

IA km

( Blak

al

mk

yes

(2)

mk

&

C

Q

113^-1=11311*1

slide-14
SLIDE 14 04-09 class page 4/20

injection

* tab

⇒ fla)#fld

f :

A → B

IAI - m

IBKK

count injections

④

lack

  • Hifk
  • mtf

N

111111

I 1111 I
slide-15
SLIDE 15 04-09 class page 5/20

"Eiffel

"

O

:# Imf

:¥

.

G) m2 19

61255

=¥

. . ..k¥

Pr

sing .)={

large if k large

Shell if k smell

slide-16
SLIDE 16 04-09 class page 6/20

I

set

52¥10

finite

"sample space " 52 cards

1521=52 !

flip n coins

HHTHTTT

2

"

probability dish

P:D → R Cal C- xEr)(PG) 203

$ ,P)

( b)

Epix)

  • I
×ER
slide-17
SLIDE 17 04-09 class page 7/20

Event

A Er

Iska

# events is 2

"

P (A) =[ Pix)

D

Pcr) =/

"A

P(01=0

empty sum

slide-18
SLIDE 18 04-09 class page 8/20

AND =D

PCAUB)=PCAHPlB)

Dc

(Union bound)

A ,

n . Am Er

P!

Ai) .si?PCAi)

slide-19
SLIDE 19 04-09 class page 9/20

RANDOM

VARIABLE

X :D → R

poker hand : 5cards

  • ut I
slide-20
SLIDE 20 04-09 class page 10/20

EXPECTED

VALUE

EIN - E Nal .PK)

GER

  • weighted average

Uniformdirt

Haen)(Plath)→ECxkEXnI

arithmetic mean

slide-21
SLIDE 21 04-09 class page 11/20

X : # heads

in a

coin flips

tri

lrayeCX7fnt@ThmECxl-Ey.P

(X -g)

DOI

YERage(x)

~

"x=g"EatsKaty's
slide-22
SLIDE 22 04-09 class page 12/20

Predicate : I → { 0,1} ←

subsets

A Er

f- *G) ={

' x EA

O x¢A

indicator function

1-

membership

slide-23
SLIDE 23 04-09 class page 13/20

ta

A Er

indicator variables ⇒ events

I

stheta

E (tf )

= I . P (if= 1)to . . . . .
  • Thr tf

't

slide-24
SLIDE 24 04-09 class page 14/20

X

, Y :D → IR

ELXTYKECXJTECY)

{

E(ax )=cECx )

  • E( Eic:X.li?ciEHi )

LINEARITY

OF EXPECTATION
slide-25
SLIDE 25 04-09 class page 15/20 HAT -CHECK n customers → n hats

fr I = n !

lucky

: got their own heh X = # lucky customers

tf

YIU Yu

slide-26
SLIDE 26 04-09 class page 16/20

Yi

: indicator that person # i n is lucky

X

= E Yi i= I E(x ) = E Ely . ) =I ply! I) = h .I =/ it -

I

slide-27
SLIDE 27 04-09 class page 17/20

Events A ,B

A trivial event :

independent

: ¥¥}

PTA NB) - PLA) . PCB)

  • H , B

always

instep

r ,B

" "
  • i. t€¥E

PCANBNC) - PCA)

  • PIB)
  • PK)

( #

A , B indep .
slide-28
SLIDE 28 04-09 class page 18/20

IDOL

A , A indep ⇐ A trivial
  • DEI

AB , Cindy if

(a)

Plan B n c) =P(A) PCBJ.PK)

(H and they are pairwise indep .

④

Cb) # Ca)

smallest

counterexample

slide-29
SLIDE 29 04-09 class page 19/20 Ck]={I, . . .,k3

DEFA

, n . idk

independent

(V-IEEKTYPi.ch/ti)=tTPlAiD

[EI
  • I=$ ?
empty product -I

21k - i

i

Air

✓

Conditions
slide-30
SLIDE 30 END WEEK 1 04-09 class page 20/20 EX Sf A ,
  • i.Am
hontiv indeed ⇒ Irl zI *
  • EX

T worthier pairwise indef

  • nlRIE2mPCAi
) =L