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

Dec 05, 2022 •107 likes •422 views

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

  1. HONORS COMBINATORICS SPRING 2020 Week 1 slides April 7–9

  2. 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

  3. htt hyetp lane in h =D 04-07 class page 2/4 - O ) ( O do . . 0,10 . - yea i - - - ↳ is I " at pairwise . - pm E IR p , equal dist ⇒ m Intl

  4. 04-07 class page 3/4 " - pm EIR pic . GEER " pi - pjHE{ sit } → I h= 11-1 m =-3 TT ÷¥¥E is this ? Max

  5. 2 - distance set " HR in 04-07 class page 4/4 tee . disease . as

  6. " 2 SEIR - distance ISI > en ' - ( I )=Nz 1st 04-08 PROBLEM SESSION Page 1/5

  7. f- I g polynomials Over IR divides : ⇐ e) ( g - f. h ) - - exponent poly prime : = 3. It 4 × 19-8 × 37 f- ( x ) - expflg ) PROVE :# f-tokzg-topn.me 04-08 problem session page 2/5

  8. A iii. i' iii. to ' = ' to 04-08 problem session Page 3/5 % .

  9. ÷i ± . Eee %÷÷÷%i÷ . 04-08 problem session page 4/5

  10. I 2 3 4 1234 ④ 3452¥ , { Lk 4 56 3 ✓ 5670 0000 4444 04-08 problem session page 5/5

  11. A- { a ,b , c } { a , b. a , c. c } = - B A if ( Hx )CxeA ← sxEB ) ( Al =3 cardinality of A 04-09 CLASS Page 1/20

  12. Akaka } B : codomain a# : # → B f 13=112 A : domain A b T z range (f) ={?fE c - it , C- codomain 04-09 class page 2/20

  13. BA - { f : A → B } IA km ( Blak mk al yes mk (2) & C Q 113^-1=11311*1 04-09 class page 3/20

  14. injection ⇒ fla ) # fld * tab A → B f : IAI - m IBKK count injections ④ - Hifk lack - mtf N 111111 I 1111 I 04-09 class page 4/20

  15. " Eiffel :# Imf :¥ " O G) m2 19 . 61255 Pr =¥ . . .k¥ . large if k large . )={ sing Shell if k smell 04-09 class page 5/20

  16. I 52¥10 set finite " " sample space 1521=52 ! 52 cards flip n coins " 2 HHTHTTT probability dish P :D → R Cal C- x Er ) ( PG ) 203 $ ,P ) ( b ) - I Epix ) × ER 04-09 class page 6/20

  17. A Er Event " Iska # events is 2 P (A) = [ Pix ) D " A Pcr ) =/ P ( 01=0 empty sum 04-09 class page 7/20

  18. AND =D PCAUB )=PCAHPlB ) ( Union bound ) Dc A , . Am Er n Ai ) .si?PCAi ) P ! 04-09 class page 8/20

  19. RANDOM VARIABLE X :D → R poker hand : 5 cards out I 04-09 class page 9/20

  20. VALUE EXPECTED EIN - E Nal .PK ) GER weighted average - Uniform dirt Haen )( Plath ) → ECxkEXnI arithmetic mean 04-09 class page 10/20

  21. coin flips X : # heads in a lrayeCX7fnt@ThmECxl-Ey.P tri ( X - g) DOI YE Rage ( x ) ~ " x=g" Eats Katy 's 04-09 class page 11/20

  22. Predicate : I → { 0,1 } subsets ← A Er f- * G) = { x EA ' O x¢A indicator function 1- membership 04-09 class page 12/20

  23. A Er ta indicator variables ⇒ events I s theta E ( tf ) = I . P ( if = 1) to . . . . . - Thr tf 't 04-09 class page 13/20

  24. , Y :D → IR X ELXTYKECXJTECY ) { E( ax )=cECx ) E( Eic :X .li?ciEHi ) - OF EXPECTATION LINEARITY 04-09 class page 14/20

  25. HAT - CHECK → n hats n customers fr I = n ! : got their own heh lucky = # lucky customers X tf YIU Yu 04-09 class page 15/20

  26. Yi : indicator that person # i is lucky . ) = I ply ! n = E Yi X i = I = E Ely E ( x ) I ) = h . I =/ it - I 04-09 class page 16/20

  27. A trivial : ¥¥} event : A ,B Events independent PTA NB ) - PLA ) . PCB ) - H , B always i. t € ¥E instep r ,B " " PCANBNC ) - PCA ) - PK ) - PIB ) ( # A , B indep . 04-09 class page 17/20

  28. IDOL indep ⇐ A trivial A , A - AB , Cindy if DEI Plan B n c) =P (A) PCBJ.PK ) (a) ( H and they are pairwise indep . ④ Cb ) # Ca ) smallest counterexample 04-09 class page 18/20

  29. Ck ]={ I , . . . ,k3 DEFA n . idk independent , ( V-IEEKTYPi.ch/ti)=tTPlAiD [ EI - I=$ ? empty product - I ✓ Air 21k - i i Conditions 04-09 class page 19/20

  30. EX hontiv indeed Sf A , - i. Am ⇒ Irl z I - * T worthier pairwise indef EX onlRIE2mPCAi ) =L 04-09 class page 20/20 END WEEK 1

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