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inversion formulae:Speaker: ͅ WCQ Date: 2005.11.28A OutlineAn overview of principle of inclusion-exclusion Principle of inclusion-exclusion Applications of PIE Mbius Inversion formula Applications of MIF(f +e'Overview of PIEOverview of PIEOverview of PIEOverview of PIE)Overview of PIE*Overview of PIE1Overview of PIE: Overview of PIE;  Principle of Inclusion-Exclusion!!(Notations: S: a set of N-elements E1,E2,& ,Er: not necessarily distinct subsets of S N(M): # of elements of S in for Nj F       &  ,*T @  Principle of Inclusion-Exclusion!!(Theorem 1 (PIE). ! Principle of Inclusion-Exclusion!!(Theorem (Purest form of PIE) Let be an n-set. Let be the 2n-dimensional vector space (over some field K) of all functions . Let be the linear transformation defined by Then exists and is given byf+" Principle of Inclusion-Exclusion!!(4Combinatorial situation involving previous Theorem 52# Principle of Inclusion-Exclusion!!(4Combinatorial situation involving previous Theorem 52$ Principle of Inclusion-Exclusion!!( Dual form  C Applications of PIEVApplication 1: Derangements How many permutations have no fixed points? i.e. BW  4K Applications of PIEApplication 1: Derangements .  PApplications of PIErApplication 2: Surjections Give X be an n-set and Y be a k-set. How many surjection of X to Y?s    YSApplications of PIE$Application 2: Surjections Note that.%     WApplications of PIEApplication 3: Show that  kApplications of PIEApplication 3: Show that  `Applications of PIEApplication 4: Euler function Let be a positive integer and be the number of integers k with such that Show that@ \)p 4Theorem 2 [Gauss] Let with . Then5 , Example We show that  Mbius Inversion formula( &Theorem 3 Let with . ' Mbius Inversion formula( <Remarks Using the Mbius function, we can reformulate as=6%Mbius Inversion formula( ]Definition An arithmetic function is a function whose domain is the set of positive integers.2^ =Mbius Inversion formula( dTheorem 4 (Mbius Inversion Formula) Let be arithmetic functions. Then if and only ifFe  B TMbius Inversion formula( %Theorem 4 (Mbius Inversion Formula) F&   Mbius Inversion formula( %Theorem 4 (Mbius Inversion Formula) F&   Applications of MIFThe Mbius Inversion Formula can be used to obtain nontrivial identities among arithmetic functions from trivial identities among arithmetic functions.Applications of MIFRExample 1 Let with and let for all such n. We have .S = Applications of MIFEExample 2 Let with and let . We have F <Applications of MIFEExample 3 Let with and let . 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Gx  'x   <A "??l "2  1C`ENH5dQ `TyS``TyS``TyS``T     Hh "`p N  71 2C f2  0"`{ # f2  0"`}-   H "`p {  71 2C   H "`p n  71 2C 2 @ 1C`ENGةH5dJ/TQ `TyS``TyS`/TyS`/T:    H "`o p ,$ 0 ;2 2G H  0޽h ?/   ̙33___PPT10.c+x D3'  = @B D' = @BA?%,( < +O%,( < +D%' =%(D' =%(D' =4@BB%BB%(D' =-g6B fade*<3<*D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*D%' d=+4 8?`CB ppt_yBCBppt_y-.03B*Y3>B ppt_y<*)?DA' =+4 8?\CB ppt_yBCBppt_y+1B*Y3>B ppt_y<*%(d)?D' =1:Bhidden*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<* %(D/' =%(D' =A@BB%BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D-' =+4 8?hCB#ppt_y+1BCB#ppt_y-.03B*Y3>B ppt_y<*)?DI' d=+4 8?dCB#ppt_y-.03BCB#ppt_yB*Y3>B ppt_y<*%()?+8+0+0 +  0  m(  x  c $    f  0vvG" m f2  0"`}- f2  0"`} &  0  `A (? ? | (  ~  BA )? ? a | )x  <A 2?? 5J  2x   <A +??l +  H "`p {  71 2C   H8 "`p n  71 2C 2  1C`ENH5dQ `TyS``TyS``TyS``T     H "`p N  71 2C H  0޽h ?  ̙33___PPT10i.c+D=' = @B +  0 0 :(  x  c $    f  0vvG" m l2  6f"`_  0  `A :? ?p R :  ~  BA ;? ? c  ;x  <A <??l <x   <A B?? B4  Bl2  6f"` - l2  6f"` 9 ~   BA >? ?   >H  0޽h ? ̙33___PPT10i.c+D=' = @B +S  0 @((  x  c $д    f  0vvG" m  0  `A C? ?p R C  ~  BA D? ? c  Dx  <A E??l Ex  <A J?? B  Jf2  0"` 9 ~   BA L? ?   Lf2  0"` 6 f2  0"` - f2  0"` + f2  0"`_ f2  0"`` F8     *  BECDE0FpAALL+: MZgq| $3B%P+]1j7w>FNWakx&/ 419B:X=i@zCDEEDCB=8.;&Qgw 58@                             B^CDEFXAA::1Ee{ #*4>HV d2xGVeo{.DYp"Fb~uaC*,9BJRxXS[<^%\[),@                       BGCDE FdAACCG8)  ziXG 7'"',3:CLU\grxk^RF:,#4 J`n 1I&^/s9CMXdp/0@                          H "`2 p ,$ 0 71 2C   H "`H. ,$ 0 71 2C   H "` w P ,$ 0 71 2C   H "`c ] ,$ 0 71 2C  ! H "`x } ,$  0 71 2C  " H "` ! | W ,$  0 71 2C  # H "`O. ,$  0 ;2 2G  $ H "` g R ,$  0 ;2 2G  % H "`w & ,$ 0 71 2C  & H "`  p Q ,$ 0 ;2 2G  ' H "`c N ,$ 0 ;2 2G  ( H4 "` j O ,$ 0 ;3 2G H  0޽h ? ̙33::___PPT10q:.c+5D6'  = @B DH6' = @BA?%,( < +O%,( < +D5' =%(D ' =%(D' =4@BB%BB%(D' =-g6B fade*<3<* D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<* D%' d=+4 8?`CB ppt_yBCBppt_y-.03B*Y3>B ppt_y<* )?DA' =+4 8?\CB ppt_yBCBppt_y+1B*Y3>B ppt_y<* %(d)?D' =1:Bhidden*o3>+B#style.visibility<* %(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D:' =%(D' =4@BB%BB%(D' =-g6B fade*<3<*D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*D%' d=+4 8?`CB ppt_yBCBppt_y-.03B*Y3>B ppt_y<*)?DA' =+4 8?\CB ppt_yBCBppt_y+1B*Y3>B ppt_y<*%(d)?D' =1:Bhidden*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-g6B fade*<3<*$D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-g6B fade*<3<*#D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-g6B fade*<3<*!D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-g6B fade*<3<*"D' =%(D' =4@BB%BB%(D' =-g6B fade*<3<*D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*D%' d=+4 8?`CB ppt_yBCBppt_y-.03B*Y3>B ppt_y<*)?DA' =+4 8?\CB ppt_yBCBppt_y+1B*Y3>B ppt_y<*%(d)?D' =1:Bhidden*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*"%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*$%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<* %(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-g6B fade*<3<*(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =-g6B fade*<3<*&D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-g6B fade*<3<*'D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-g6B fade*<3<*%++0+0 ++0+ ++0+0 ++0+0 ++0+0 ++0+ 0 ++0+ 0 ++0+!0 ++0+"0 ++0+"0 ++0+#0 ++0+$0 ++0+$0 ++0+%0 ++0+&0 ++0+'0 ++0+(0 +Q  0 ` !(  x  c $p9   9  f  0vvG" m  0  `A R? ?p R R 9 ~  BA S? ? c  Sx  <A T??l Tx  <A U?? B  Uf2  0"` 9 ~   BA V? ?   Vf2  0"` 6 f2  0"` - f2  0"` + f2  0"`_ f2  0"`` fF      0  BECDE0FpLL+: MZgq| $3B%P+]1j7w>FNWakx&/ 419B:X=i@zCDEEDCB=8.;&Qgw 58@                             B^CDEFX::1Ee{ #*4>HV d2xGVeo{.DYp"Fb~uaC*,9BJRxXS[<^%\[),@                       BGCDE FdCCG8)  ziXG 7'"',3:CLU\grxk^RF:,#4 J`n 1I&^/s9CMXdp/0@                          H9 "`2 p  71 2C   H09 "`H. ,$ 0 71 2C   H #9 "` w P ,$  0 71 2C   H'9 "`c ] ,$  0 71 2C   H*9 "`x }  71 2C   H.9 "` ! | W ,$ 0 71 2C   Hh29 "`G*  ;2 2G   H`69 "` k J ,$ 0 72 2C   H :9 "`w &  71 2C   H49 "`  t Y  ;2 2G   HXA9 "`g Z  ;2 2G   H,E9 "` d N  ;3 2G   HLI9 "` f A ,$  0 ;0 2G x ! <A Z??z !  ZH  0޽h ? ̙33P8H8___PPT10(8.c+D\5' U9 = @B D5' = @BA?%,( < +O%,( < +D: ' =%(D ' =%(D' =A@BB%BB0B%(D' =-g6B fade*<3<*D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*D%' d=+4 8?`CB ppt_yBCBppt_y-.03B*Y3>B ppt_y<*)?DA' =+4 8?\CB ppt_yBCBppt_y+1B*Y3>B ppt_y<*%(d)?D' =1:Bhidden*o3>+B#style.visibility<*%(D' =A@BB%BB0B%(D' =-g6B fade*<3<*D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*D%' d=+4 8?`CB ppt_yBCBppt_y-.03B*Y3>B ppt_y<*)?DA' =+4 8?\CB ppt_yBCBppt_y+1B*Y3>B ppt_y<*%(d)?D' =1:Bhidden*o3>+B#style.visibility<*%(D(' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D: ' =%(D ' =%(D' =A@BB%BB0B%(D' =-g6B fade*<3<*D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*D%' d=+4 8?`CB ppt_yBCBppt_y-.03B*Y3>B ppt_y<*)?DA' =+4 8?\CB ppt_yBCBppt_y+1B*Y3>B ppt_y<*%(d)?D' =1:Bhidden*o3>+B#style.visibility<*%(D' =A@BB%BB0B%(D' =-g6B fade*<3<*D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*D%' d=+4 8?`CB ppt_yBCBppt_y-.03B*Y3>B ppt_y<*)?DA' =+4 8?\CB ppt_yBCBppt_y+1B*Y3>B ppt_y<*%(d)?D' =1:Bhidden*o3>+B#style.visibility<*%(D(' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D: ' =%(D ' =%(D' =A@BB%BB0B%(D' =-g6B fade*<3<*D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*D%' d=+4 8?`CB ppt_yBCBppt_y-.03B*Y3>B ppt_y<*)?DA' =+4 8?\CB ppt_yBCBppt_y+1B*Y3>B ppt_y<*%(d)?D' =1:Bhidden*o3>+B#style.visibility<*%(D' =A@BB%BB0B%(D' =-g6B fade*<3<*D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*D%' d=+4 8?`CB ppt_yBCBppt_y-.03B*Y3>B ppt_y<*)?DA' =+4 8?\CB ppt_yBCBppt_y+1B*Y3>B ppt_y<*%(d)?D' =1:Bhidden*o3>+B#style.visibility<*%(D(' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+ 0 +Y   0 p$R(  x  c $j9   9  f  0vvG" m  0  `A [? ?p R [ 9 ~  BA \? ? c  \x  <A ]??l ]f2  0"` 9 ~   BA _? ?   _f2  0"` 6 f2  0"` - f2  0"` + f2  0"`_ f2  0"`` fF      0  BECDE0FpLL+: MZgq| $3B%P+]1j7w>FNWakx&/ 419B:X=i@zCDEEDCB=8.;&Qgw 58@                             B^CDEFX::1Ee{ #*4>HV d2xGVeo{.DYp"Fb~uaC*,9BJRxXS[<^%\[),@                       BGCDE FdCCG8)  ziXG 7'"',3:CLU\grxk^RF:,#4 J`n 1I&^/s9CMXdp/0@                          H q9 "`2 p  71 2C   Hdu9 "`H.  71 2C   H4y9 "` w P  ;0 2G   H(}9 "`c ]  71 2C   H9 "`x }  71 2C   H9 "` ! | W  71 2C   HT9 "`w &  71 2C  " H9 "` m H ,$ 0 71 2C ~  BA ^?? B  ^~ ! BA a??z -  a~ # BA f??j  fH  0޽h ? ̙333 + ___PPT10 .c+ RDo ' 9 = @B D* ' = @BA?%,( < +O%,( < +Dq' =%(D' =%(D' =A@BB%BB0B%(D' =-g6B fade*<3<*D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*D%' d=+4 8?`CB ppt_yBCBppt_y-.03B*Y3>B ppt_y<*)?DA' =+4 8?\CB ppt_yBCBppt_y+1B*Y3>B ppt_y<*%(d)?D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-g6B fade*<3<*"+p+0+0 ++0+"0 +$  0  $(  r  S T9   9  r  S 9 PF  9  x  <A h??  hx   <A n??N  nx   <A v?? vx   <A y??V f^ yH  0޽h ? ̙3380___PPT10.   0 4 ,  (  x  c $(9   9  x  c $9 P 9  x   <A s??: s  09 8j4X 6Proof: 28 X$  X$ @ X$  X$    0P9 X$  W'Case1. and is in none of ( 2(h  c $Af ??88 h  c $A_ ??-Y h  c $Ai ?? (g $  `  c $A_ ??g ?    09  ,  d4contributes 1 to left-hand side and right-hand side.5 25x8  :   :   09  .  x6Case2. and is in exactly k of the sets &7 2%h  c $Af ??,B   h  c $A_ ??!i   h  c $Ag ??!> :  `  c $A_ ??_ 3  ;8  n   n   0,9  H$  Kcontributes 0 to left-hand. 2  09  n  `0The contribution of to right-hand side equals1 21h  c $A_ ??3 U  `  c $Ah ??h ` H  0޽h ? ̙3380___PPT10.  0 xp p(  px p c $9   9  x p c $9 P  9  `  p c $Ar ??: ` p c $A ??hX, ` p c $Aq ??nK 2 ` p c $AV ??62 ` p c $Au ??<`  ` p c $As ?? B0  ` p c $At ??   H p 0޽h ? ̙3380___PPT10.   0 d \ t(  tx t c $ 9   9  x t c $9 Pv 9  ` t c $Ao ??ia   t 09 p$ ; : given set 2 `  t c $A ??|\@   t 09 Hzh @: properties set 2` t c $Av ??gV  t 09 ,nL < : subset of  2 ` t c $A ??`$ ` t c $Aw ??T   t 09   z8: # of objects in that have exactly the properties in&9 2` t c $Av ??3 ` t c $Ao ??-  ` t c $Ax ??$ N   t 0N   {9: # of objects in that have at least the properties in&: 2` t c $Ao ??/   ` t c $Av ??;   H t 0޽h ? ̙3380___PPT10.^  0  x^(  xx x c $ N   N  x x c $ N Pv N  ` x c $Ak ??_    x 0N p: CBy previous Theorem 2 x 0tN X^Dx < Then we have 2  x 0N  dJ  ?In particular,  2` x c $Ay ?? ~ ` x c $A| ?? e  H x 0޽h ? ̙3380___PPT10.  0 |(  |x | c $N   N  x | c $N Pv N  ` | c $A~ ??w    | 0 N  ^D2  < Then we have 2  | 0%N RdlX QWe can reformulate the PIE by interchanging with , with , and so on.R 2R` | c $A} ??   `  | c $A ?? e `  | c $Az ??N   | 0|*N t z8: # of objects in that have at most the properties in&9 2`  | c $Av ??e `  | c $Ao ?? _ ` | c $A ??@4 ` | c $A ??V @ ` | c $A ??wB ` | c $A ??;, H | 0޽h ? ̙3380___PPT10.-  0 -(  x  c $P7N   N  x  c $8N P\ N    09N ,RL 4Sol: 2x  <A ??p L x   <Ap ??F R k8         0>N   3Let 2  <A ??   o8  P    P    0CN  @  7We have 2  <A ?? P   o8  ?  ?  0\HN   7By PIE, 2  <A ?? ? H  0޽h ? ̙3380___PPT10.j  0   j(  x  c $RN   N  x  c $SN P N  x  <A ??   x   <Al ??eJ   6 VN ]m o?Since , is a good approximation of @ @x  <Aj ??'xw x  <A ??O    6[N \o| 4From x  <Am ??1 N q    60NN  u  Mit is not difficult to derive x  <An ?? \"  H  0޽h ? ̙3380___PPT10.  0  0 (  x  c $PiN   N  x  c $(jN P\ N    0kN ,RL 4Sol: 2x  <A ??^% k O   0dN   3Let 2x  <A ??6     0 n then8    BN @[o`  a if k = n then8    6tN  IA0  9 Note that  x   <A ??   !   6lN  I 3where S(n,k) is the Stirling number of second kind.J4 )H   0޽h ? ̙3380___PPT10.   0 C ; P $(  $x $ c $xN   N  x $ c $N P N  x $ <A" ??n  $ BĠN ['   DCombinatorial proof:  $ 60N  E8  y7Let be an n-set of blue balls&8 $x  $ <A! ??Q i     $ 6dN  ?  {-How many k-subsets consist of red balls only?2.  $  $ BN ,  L  Ybe an m-set of red balls.$  $ 6@N  ?  \,The answer is trivially the right-hand side.- -p  $ 6|N  ? The left-hand side is letting S be all the k-subset of X*"Y and Ei those k-subset that contain bi and then PIE gives the left-hand side.   &?GH $ 0޽h ? ̙3380___PPT10.S   0   p,S (  ,x , c $\N   N  x , c $ N P N  x , <A" (??n ( , BN [>  >Algebra proof:  , 6|N   EWe use the expansion: x  , <A' ,??   ,8    ,    , 6dN  ,  Z*The coefficient of in is + + , <A- /??( h   / , <A. 2??e , { /  2 , <A/ 5??e   58 m s ,  , 6PN   b2The coefficient of in is 3 3 , <A3 8??  8 , <A1 ;??`  ; , <A2 >??m s > , 6xN  b2The coefficient of in is 3 3x , <A4 A?? ^  Ax , <A0 D??o l  Dx , <A5 G??n GH , 0޽h ? ̙3380___PPT10.   0 3 + `((  (x ( c $(N   N  x ( c $N P N  x ( <A, '??   ' ( BN [  6Proof: x ( <A#  ?? > W  x ( <A* ??FHX x ( <A% ??bjI x ( <A& ??Zj $ x ( <A( ??Fo- 8 m h ( mh ( 6N    3Let  ( <A+ ??   ( 6@N m   7We have  ( <A ??    ( 6D| mt  7By PIE,  ( <A$ ??bh H ( 0޽h ? ̙3380___PPT10.   0   0 (  0r 0 S |   |  r 0 S  | P> |  x 0 <A6 K??W> Kx 0 <A8 N??ds- Nx  0 <AO S??zV *  S  0 B` | [  6Proof: +8 !V 0!V 0 <A= i?? m  i  0 <| !u  4Let    0 BA9 Y??c&  Y 0 <| !  4Let   0 BA; ^??N  ^ 0 <| '   6Since   0 BA: c?? I  c 0 BA< f??  f 0 < | - e  @Since every integer from 1 to n belongs only one , we have &A " 0 BA> l??   l 0 < | 3 k /Since d runs all positive divisor of n, we have80   0 BA@ q??V qH 0 0޽h ? ̙3380___PPT10.H@{U  0 <U(  <x < c $*|   |  x < c $+| P> |  x < <AB ??B    < 6-| rg1 7We have x  < <A) ??  x < <AZ ?? QT H < 0޽h ? ̙3380___PPT10.H@{   0 #  98 (  8 '8 6<7|   vMbius Inversion formula(   (8 0X;|  P> RDefinition Let with . The Mbius function, denoted by isS I+"x )8 <A6 ??c> x *8 <A8 ??ds- x +8 <AA ??teZ  ,8 6A|   B , if [F   ]  -8  ]  .8 <AE ??s   /8 6DE|  , X1, if n =1&  08 <AF ??  `r 18 0LT 28 6\J|  A0, if  38 <AG ??|   48 <AY ?? &]  F a   58  a N a   68 a    78 6 |  x @ <A6 ??c> x @ <A8 ??ds- x @ <AC ??P   @ <Y| > S1, if n =1&  ^r  @ 6~L  @ <X_|  C0, otherwise  @ B|c| [  6Proof: K8   _  @ _  @ 6@g|   _  CThere is nothing to show when n=1. If , then&D % @ <A* ?? p   @ <AK ?? p  H @ 0޽h ? ̙3380___PPT10.H@{0  0 0\0(  \x \ c $4q|   |  x \ c $ r| P  |  x \ <A[ ??" W x \ <A? ??0N3X H \ 0޽h ? ̙3380___PPT10.H@{i  0 Hi(  Hx H c $`}|   |  x H c $8~| P> |   H 0|u|  sma  Q Example (x H <A7 ??  x H <A\ ?? z 1  x H <A^ ??5 r  H H 0޽h ? ̙3380___PPT10.H@{  0  L(  Lx L c $|   |  x L c $l| P  |  x L <AL ??b H x L <AM ?? = x L <AN ??   H L 0޽h ? ̙3380___PPT10.H@{H  0  PH(  Px P c $|   |  x P c $| P  |  x P <AH L??S< Lx P <AS c??dp c`8 $(  P$(  P <A{  ??$(}   @ ; W  P  P 6ܐ| ; W Jthe inner sum is 0 unless    P <AR y??  y  P B̕| P[p 6Proof: H P 0޽h ? ̙3380___PPT10.H@{  0 h` T(  Tx T c $8|   |  x T c $| P  |  x T <AH d??S< dx T <AP k??dp k8 ;v   Tv;  T 6| ; W  Jthe inner sum is 0 unless   T <AT  ??vd    T <AU q??  q  T B| P[p 6Proof: H T 0޽h ? ̙3380___PPT10.H@{  0  @`(  `r ` S |   |   ` s T| 0e0e #" 0ePp |  H ` 0޽h ? ̙3380___PPT10.[0mԸ  0   X(  Xx X c $|   |  x X c $| P  |  x X <AX ?? k  x  X <AW ??`  : x  X <A6 ??}i8 x  X <A8 ??^"' x X <AQ ??%  X 6h| F[f  Y)By MIF, we obtain the nontrivial identity* * X 6|  %  Aor , equivalently x X <A] ??   H X 0޽h ? ̙3380___PPT10.H@{w  0 P dw(  dx d c $|   |  x d c $l| P  |  x d <A` ??   x d <A6 ??}i8 x d <A8 ??^"' x d <Ab ?? 4%   d 6| F[f  Y)By MIF, we obtain the nontrivial identity* *x  d <Aa ??l L F H d 0޽h ? ̙3380___PPT10.H@{w  0 ` hw(  hx h c $|   |  x h c $| P  |  x h <Ad ?? o  x h <A6 ??}i8 x h <A8 ??^"' x h <Ac ??`%  h 6 | F[f  Y)By MIF, we obtain the nontrivial identity* *x  h <Ae ??l Z F H h 0޽h ? ̙3380___PPT10.H@{(  0 ?7p l(  lx l c $P|   |  x l c $| P  |    l 0| 3 s  7-END- 2<H l 0޽h ? ̙33___PPT10i.H@{+D=' = @B +xKLcUW"Z3A4n&P 6ĤV}hB3 ƥq 'qq}Qg$n}&&.&c<Ϲ s9s=w>#3}}iMok?6}޿[?^ғ߂ T@:":9rzƿ7qF{X UߥJ ΰ%]SrژúlATI^A;H_C\GU?3Qi$3W犯as~ki%}QO#=]5eB4_^/fֆ5J1QORqsG;L^gS t+t3>;ElH)]2j3sOt].sL!De=yE/@$X{w$0[\|1írV_NBif.Qz|*(@'Ćc۱sx:p*r Mi¡ яmȋ!iF~_z2p'` 0J  B2J--6GwLS=?2aށ,&*~tXX?:apTVEhA2Rp XзPK2̩5h6?C#k+'qΑ*y' 8 ijMaUԂ~Qw'̵/޽&: :%3㶗;fl5/q.ܹ>cʧv/!n}A;&J(RoD `[K xރjcA{%/$Y>A ɺȱJGs3 d3‡@=25)hY1h!&CNgf_yp_}fGvXJ~:= xXkA~o6MMVkzbnCchRDbUI HxwxP^!j@C7;m")hcf̼ɛ7oM_}V(PUC&>j5RHj-4TKn,=PA 9( npvTmF_k4k +!fxX_;tR0u!z_{H' Xuf!u^`9Pr9%H5 ӐOR2պbkWԿÇf)\9y¿ NϯÊxtxH6l"x[y'^nOÉdcщL(IgGH₞3Ly5_U'DJ>MQhv:s(1UV%:'o,-|Nh ă#Pҿ=16{C`h.c~$N5wJzN )H=j5B^ #H״Bm]H捹a"{yց7'R\S(Sw{Vr0אDQۃ[-]<0+c\^賘&,2L$%2L<0(\/V{=cg2g[ӝ gIEeh <.Ѵ]q0m66dE5"(ҿ)<+A^s}xc>GwDqa0#j< F¡%B^xXOA3KJ"*!c)z6ċI]t5vQzЃ&Ą`LHfO[@S ׾̛v{ok_w~  Vh.$*>+J%ƒJ #w!5C?Ƞ@,+CсQ\9E㜢ZF=믥Yn>Xb-3n>yZX!<,ːނ(vAS01 w@,dI^w^ sT~v"\h[](0 3iuבWx9=>t߱D=wzX `V`1 Ƀi+F?tq}obyz#τćuŇ0g^ӡdrXÙZT)-'G{x&/L™e17e徑U2orK^ml`.m ,$ P}45xC`* sq857E[~Qp#/ z/Ew&Fɾ!&K$%l7asd02Q0#?,H%w,JB/D8 ,?o&xAK.*A@qJ}.R?c&/cֿyd@PA[PK&? 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Principle of Inclusion-Exclusion!!(Theorem (Purest form of PIE) Let be an n-set. Let be the 2n-dimensional vector space (over some field K) of all functions . Let be the linear transformation defined by Then exists and is given byf+" Principle of Inclusion-Exclusion!!(4Combinatorial situation involving previous Theorem 52# Principle of Inclusion-Exclusion!!(4Combinatorial situation involving previous Theorem 52$ Principle of Inclusion-Exclusion!!( Dual form  C Applications of PIEVApplication 1: Derangements How many permutations have no fixed points? i.e. BW  4K Applications of PIEApplication 1: Derangements .  PApplications of PIErApplication 2: Surjections Give X be an n-set and Y be a k-set. 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Then if and only ifFe  B TMbius Inversion formula( %Theorem 4 (Mbius Inversion Formula) F&   Mbius Inversion formula( %Theorem 4 (Mbius Inversion Formula) F&   Applications of MIFThe Mbius Inversion Formula can be used to obtain nontrivial identities among arithmetic functions from trivial identities among arithmetic functions.Applications of MIFRExample 1 Let with and let for all such n. We have .S = Applications of MIFEExample 2 Let with and let . We have F <Applications of MIFEExample 3 Let with and let . 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