ࡱ> XZIJKLMNOPQRSTUVWq`0sbjbjqPqP6::&>>>>####8#_VV"xxx<<<[[[[[[[$ehKh@[#6Ԓd866[>>xx\^"""6>xj#8x["6[""+L#4xJ pU.[ʶ.\: ^0_D/h^`h4h#4< "&ʱl<<<[[d<<<_6666#D$$Ĩ#$$###>>>>>>      1J'6J'* 'D5A 'D#HD 'D+'FHJ 'DA5D 'D/1'3J 'D#HD   'D('( 'D#HD 'DEF7B 'D1J'6J H 'DE,EH9'*  'DEF7B 'D1J'6J  'D,ED) %F4'&J) .(1J) GJ 'D,ED) 'D*J D' *-ED .(1 GJ 'D*J *-ED .(1' #H #C+1 HJECF 'DFGJ  'D#E1  'DF/'!  'D*9,(  'D7D(  'D*EFJ 'D-CE 9DJG' ('D5H'( #H 'D.7'!  [1] 'D9('1) 'D1J'6J) : 'D9('1) GJ CD ,ED) .(1J) *-ED .(1' #H #C+1 5H'( #H .7'! . E+'D (1) : 'C*( +D' + ,ED %F4'&J) . (1)....................................... (2)....................................... (3)............................... 'C*( +D'+ ,ED %.('1J) . (1)....................................... (2)....................................... (3)............................... [2] #FH'9 'D9('1) :  1- 'D9('1) 'D(3J7) : GJ 'D9('1) 'D*J *-ED .(1' H'-/ AB7 5H'( #H .7'! . 2- 'D9('1) 'DE1C() : GJ 'D9('1) 'D*J *-ED #C+1 EF .(1 5H'( #H .7'! . 3- 'D9('1) 'DEA*H-) :GJ CD 9('1) *-HJ E,GHD #H #C+1 . E+'D (2) : 'C*( 9('1) H'-/G 9DI 'D#BD EF CD FH9 . (1)................................................................................................................... (2)................................................................................................................... (3)..................................................................................................................  ,/'HD 'D5/B [3] ,/HD 'D5/B : ( : 9('1) *CHF 5'&() #H .'7&)  ( ( : FAJ 'D9('1) ( . E+'D (3) : '0C1 FH9 CD EF 'D9('1'* 'D*'DJ) : %F 'DDG J-( 'DE-3FJF ................... 'D4E3 *41B EF 'D:1( .............. 3 + 3 < 7 H E3'-) 'DE1(9 = 4 7HD 6D9G ................................. E+'D (4) : ( : (J1H* 9'5E) 'D,2'&1 . ( ( : ...................... [4] #/H'* 'D1(7 : #/') 'D1(7H#H%0' ......A%F%0' H AB7 %0' 'D1E2(((( *91JA : *CHF 'D9('1) ( ( ( 5'&() AJ -'D) H'-/G AB7 . 9F/E' JCHF ( 5'&(G  ( 5'&(G #J6'K . *CHF 'D9('1) ( ( ( .'7&) AJ -'D) H'-/G AB7 . 9F/E' JCHF ( .'7&)  ( .'7&) #J6'K . *CHF 'D9('1) ( ( ( .'7&) AJ -'D) H'-/G AB7 . 9F/E' JCHF ( 5'&(G  ( .'7&) . *CHF 'D9('1) ( ( ( 5'&() 9F/E' JCHF (  ( DGE' FA3 BJE 'D5H'( .( 5'&(*JF E9'K #H .'7&*JF E9'K ) */1J( : #CED ,/HD 'D5/B . ((( ( ( ( ( ( ( ( ( ( ( ( E+'D (5) : %0' C'F* (: *9FJ E,EH9 2H'J' 'DE+D+ 180 R  ( : *9FJ 3 + 7 = 12 . 9JF BJE 'D5/B DE' JDJ: (1) ( ( ( = ............................ (2) ( ( ( = ......................... (3) ( ( ( = .......................... (4) ( ( ( = .......................... E+'D (6) : 'CED ,/'HD 'D5/B DD9('1'* 'D*'DJ) (1) ( ( ( )( ( ( ) (2) ( ( ( (  ((( (( ( ( ( ( ( )( ( ( )   (*0C11111111111111 : D#I 9('1*JF (  ( . A%F (1) ( ( ( ( ( ( ( (2) ( ( ( ( ( ( ( (3) ( ( ( ( ( ( ( #I #F CD EF 'D1H'(7 ( (  (  ( ) *-BB .'5J) 'D#(/'D */1J( : 69 'D1'(7 'DEF'3( AJ 'DA1': : ( JCHF 'D4CD /'&1) ..........C'F* CD FB7) AJG *(9/ (9/'K +'(*'K 9F FB7) E1C2J) AJG . ( *CHF 'D9// 5-J-' ....... DE J-HJ 9DI C31 ( JCHF 'D9// 2H,J' ......... B(D 'DB3E) 9DI 2 .  [5] 'D9('1'* 'DEC'A&) :  FBHD %F 'D9('1*JF (  ( E*C'A&*'F %0' C'F DGE' BJE 'D5/B FA3G' . H*C*( ( ( ( . E+'D (7) : ('3*./'E ,/HD 'D5/B #+(* #F ( (( ( ( ) ( ( ( ( ( ( ((( ( ( ( (( ( ( )( (( (( ( ( ( ( [6] F81J) : (  ( #J 9('1*JF (1) ( ( ( (( ( ) . ( ( ( ( ( ( ( ( ( ( ( ( ( ( . ( (( ( ( ) ( ( ( ( ( . ( (( ( ( ) ( ( ( ( ( ( . ( (( ( ( ) ( ( ( ( ( ( . 'D(1G'F : 1',9 'DC*'( 5 25 + 26 . E+'D (8) : #FA 'D9('1'* 'D*'DJ) (71JB*JF : 1- ( 3#2H1C 'DJHE #H '*5D (C G'*AJ' ) . 2- ( F2D 'DE71 HN '.61* 'D#16 ) . E+'D (9) : #+(* #F ( ( ( ( ( ( ( ( ( ('3*./'E ,/HD 'D5/B .  [7] 'D#B*6@@'! :  1- D#J 9('1*JF (  ( FBHD 'F ( *B*6J ( %0' C'F* 'D9('1) ( ( ( 5'&() /'&E' . H*C*( ( ( ( . 2- D#J 9('1*JF (  ( FBHD 'F ( *C'AJ! ( %0' C'F* ( ( (  ( ( ( . H*C*( ( ( ( . E+'D (10) : F'B4 5-) 'D9('1) %0' C'F* 'D9('1) :  *9FJ : 3= 3 H 'D9('1) ( *9FJ : 5 3 + 2 = 17 A%F : ( ( ( . E+'D (11) : F'B4 5-) 'D9('1) %0' C'F* 'D9('1) :  *9FJ : 3= 4 H 'D9('1) ( *9FJ : 32 + 1 = 17 A%F : ( ( ( .  [8] 71'&B 'D(1G'F :  1- 'D(1G'F 'DE('41 . 2- 'D(1G'F (%97'! E+'D E9'C3 . ( 1',9 'DC*'( 5 32 ) ( 1',9 'DC*'( 5 33 ) E+'D (12) : [1] '3*./E 'D(1G'F 'DE('41 D%+('* %0' C'F 23 + 5 = 19 A%F 3 = 7 [2] (%97'! E+'D E9'C3 F'B4 5-) CD EF 'D9('1'* 'D*'DJ) : (1) ( 3 + 5 )2 = 32 + 52 . (2) [32/ /+/ /5//2/ = 3 + 5  'DE,EH9@@@@@@@@'* Sets *EGJ/ : 3(B /1'3) F81J) 'DE,EH9'* . #C*( 'DE,EH9'* 'D*'DJG (0C1 9F'51G' : ( E,EH9) #1B'E 'D9// 35457239 = ....................................................................... ( E,EH9) BH'3E 'D9// 18 = ................................................................................. #C*( 'DE,EH9'* 'D*'DJG ('D.'5J) 'DEEJ2) D9F'51G' : ( { 9  16  25  36  49  64 } = ................................................................... ( { #  -  /  E } = ...................................................................................... [1] *9@@@@@@@'1JA : 1- ( ( ( = { ( : ( ( ( ( ( ( ( } . 'D*B'79 Intersection 2- ( ( ( = { ( : ( ( ( ( ( ( ( } . 'D#*-'/ Union 3- ( - ( = { ( : ( ( ( ( ( ( ( } .'DA1B Difference 4- ( N = { ( : ( ( ( ( ( ( ( } . 'DECED) The Complement E+'D (13) : ( = { 2  5  8  9  10  12 }  ( = { 5  9  10  12 }  ( = { 8  9  12 } . ( ( ( ( = .......................................................................................... ( ( ( ( = .......................................................................................... ( ( - ( = ...................................................................................... ( ( N = ................................................................................................ ( ( ( ( = .......................................................................................... ( ( ( ( = ......................................................................................... ( ( ( ( = ........................................................................................ ( ( ( ( = .......................................................................................... ( ( ( ( N =......................................................................................... E+'D (14) : %0' C'F* ( = { 1  2  3  4  5  6  7 } (= { 2  4  5  7 }  ( = { 1 2  3  4  5 } #H,/ E'JDJ (1) ( ( ( = & & & ................................................................... (2) ( - ( =................................................................................ (3 ) ( ( ( ( ) N = ............................................................................ ( 4 ) (N ( (N = & & & ................................................................... E+'D (15) : '.*1 'D1E2 'DEF'3( EF (JF 'D#BH'3 H69G AJ 'DA1': 'DEF'3( [ (  (  (  = ] (......{(  / }  2 & & . { 12  25 }  4 & & . { 9  4 }  { 7 } ... { 3  5  7 }  *0C1 ( ( ( ( A%F : ( ( ( =(( ( ( =(( ( ( =(( ( ( = (( ( ( = (( ( ( =(( ( ( = ( (( ( ( ( = ( ( (( ( ( N =(( ( ( N =((( ( ( ) ( ((( ( ( ) ( (( ( (( ( ( )( ( (( ( ( )( N = (( N = (  ((N ) N= (  [2] ,/HD 'D#F*E'! : EF 'D*9'1JA 'D3'(B) #CED 'D,/HD ((( ( ( ( ( ( ( ( - (((((((( [3] F8@1J) :  D#J (  ( E,EH9'* ,2&J) EF ( : 1- (( ( ( ) N = ( N ( ( N  2- ( ( ( ( ) N = ( N ( ( N . 'D(1G'F : ((  [4] F8@1J@) : D#J +D'+ E,EH9'* (  (  ( JCHF : 1- ( ( ( ( ( ( ) = (( ( ( ) ( (( ( ( ) . 2- ( ( ( ( ( ( ) = (( ( ( ) ( (( ( ( ) . 'D(1G'F : 1',9 'DC*'( 5 45 .  '.*(@@@@@@@@'1 0'*@J (1) 'D3$'D 'D#HD : '.*1 'D#,'() 'D5-J-) AJE' JDJ : 31 : %0' C'F*  : *9FJ 'D7'D( E,*G/  ( : *9FJ 'D7'D( 0CJ A'D9('1) 'DDA8J) DD9('1) 'DE1C() (   ( G@J .................... (1) 'D7'D( C3HD H0CJ (2) 'D7'D( E,*G/ HDJ3 0CJ (3) 'D7'D( DJ3 E,*G/ H0CJ (4) 'D7'D( DJ3 E,*G/ HDJ3 0CJ 32 : %0' C'F@@@@* 'D9@@('1)  : *9FJ 3 7= 21 H'D9@@@('1) ( : *9FJ ,/) EJF'! 9DI 'D.DJ, 'D91(J A%F 'D9('1) 'D5'&() /'&E'K EF 'D9('1'* 'DE1C() 'D*'DJ) GJ ................... (1)  (  ( (2)   ( ( (3)    ( (4)   ( 33 : %0' C'F*  = { 1  2  3  4  5  6  7  8 }   = { 1  2  3  4 }   = { 5  6 }  = { 2  4  6  8 } A'D9('1) 'D5'&() GJ ................. (1)   = { 2  4 } (2)    =  (3)    =  (4)  = { 1  3  5 } 34 : 'D9('1) 'D.'7&) GJ .................. (1) {2} ( {2  3} (2) {2  3}  {4} = {2  3  4} (3)   {  (} (4) 5A1   35 : %0' C'F*  = {1  2  3  4  5  6  7  8 }   = { 1  2  3  4 }   = { 5  6 }   = { 2  4  6  8 } A%F 'D9('1) 'D5-J-) GJ .................  (1)    =  (2)    =  (3) (    ) = { 6 } (4)    = { 5  6 } 'D3$'D 'D+'FJ : 31 : F'B4 5-) 'D9('1) 3 3 - 1 = 11 ( 3 = 4 32 : (%97'! E+'D E9'C3 F'B4 5-) 'D9('1) ( 3 - 5 )2 = 32 + 25 33 : %0' C'F* ( = { 4  5  6  7  8  9 } (= { 7  4  5 }  ( = { 9  7  4  8 } #H,/ E'JDJ (1) ( ( ( = ............................................................................ (2) ( - ( =.............................................................................. (3 ) ( N = ..........................................................& & & & & (4) (( ( ( ) N = .............................................................................. 34 : #CED BJE 'D5/B 'D*J *CED 'D,/HD  35 : ('3*./'E ,/HD 'D5/B #+(* #F ( ( ( ( ( ( ( ( ( ) ( ( ( ( ( ( ) . EG@@@@@@@'1'* *E1JF (1) : 8DD 'DE,EH9) 'DEH6-) '3AD 'D4CD ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (  ( ( ( ( ( ( ( ( *E1JF (2) : (  ( E,EH9*'F . %0' C'F 9// 9F'51 ( = 15 9F51  9// 9F'51 ( = 23 9F51  9// 9F'51 (( )" ( ) = 6 . A#F : (1) 9// 9F'51 (( U ( ) = .......................... (2) 9// 9F'51 (( - ( ) = .................. *E1JF (3) : (  ( E,EH9*'F . %0' C'F 9// 9F'51 ( = 25 9F51  9// 9F'51 ( = 43 9F51  9// 9F'51 (( U ( ) = 56 . A#F : (1) 9// 9F'51 (( U ( ) = ...................... (2) 9// 9F'51 (( - ( ) = ........................ 'D('( 'D+'FJ 'D9D'B'* H 'D*7(JB'*  & Mapping Relations 'D*7(JB'* Mappings *EGJ/ : E,EH9'* 'D#9/'/ 'D*J 3(B /1'3*G' GJ : 7 = E,EH9) 'D#9/'/ 'D7(J9J) Natural Numbers= ................................................................ C = E,EH9) 'D#9/'/ 'DCDJ) KWhole Numbers = ................................................................. ( = E,EH9) 'D#9/'/ 'D5-J-) Integer Numbers = ............................................................ ( = E,EH9) 'D#9/'/ 'DF3(J) Rational Numbers = ..............................................................  (= E,EH9) 'D#9/'/ 'D-BJBJ) Real Numbers = .................................................................... 7 ( C ( ( ( ( ( ( 'D9@@@D'B@) : Relation ( (  E,'D 'D9D'B) = ..................................... 'DE,'D 'DEB'(D = ...................................... 3 7 (J'F 'D9D'B) = ........................................ 4 11 E/I 'D9D'B) = .......................................... 5 8 13 */1J( : ( = { 2  5  8 }  ( = { 3  6 } . #H,/ : ( ( = ............................................................................. ( ( = ............................................................................. *0C@@@@@@1 : ( 3  5 ) ( ( 5  3 ) . ( 3  5 ) = ( (  ( ) ( 3 = (  5 = ( . */1J( : #H,/ BJE) CD EF 3  5 %0' C'F ( 3 + 3  11 ) = ( 1  5  2 5 ) .................................................................................................................. ...................................................................................................................  [1] EAGHE 'D*7(JB : F3EJ 'D9D'B) EF 'DE,EH9) ( 'DI 'DE,EH9) ( *7(JB %0' 'B*1F CD 9F51 AJ ( (9F51 H'-/ AB7 AJ ( .  *C*( ( : ( ( ( . #H ( ( ( . E+'D (1) : %0' C'F*  = {   (  , }   = { 2  4 } A%F 'D9D'B) 'D*J *E+D *7(JB@K' EF    G@@J :   E+'D (2) : ( : ( ! ( *7(JB E91A ('DB'9/) ( (3) = 3 + 2 . ',( 9E' JDJ : (1) '13E E.77' 3GEJ' :J1 C'ED D*7(JB ( EH6-' 9DJG 5H1 'D#9/'/ -2  -1  0  1  2 . (2) #H,/ ( ( 7 ) =0000000000000000  ( ( - 7 ) = 00000000000 ( ( F ) = -11 . ,/ BJE) F . #C*( E/I 'D9D'B) = ................................................................................  [2] #FH'9 'D*7(JB : 'D*7(JB ( : ( ( ( J3EI : *7(JB 4'ED %0' C'F CD 9F51 AJ ( 5H1) D9F51 H'-/ 9DI 'D#BD AJ ( . ( E/I 'D*7(JB = ( .  *7(JB *('JF %0' C'F #J 9F51 AJ ( 5H1) D9F51 H'-/ 9DI 'D#C+1 AJ ( ( 'D9F'51 'DE.*DA) EF ( DG' 5H1 E.*DA) EF ( ) ( ( ( 31) = ( ( 32) ( 31 = 32 *7(JB *B'(D %0' C'F *('JF H4'ED .  *('JF 4'ED *B'(D E+'D (3) : ( : ( ! ( -J+ ( ( 3 ) = 3 3 1- #H,/ ( ( 6 ) =................ ( ( - 5 ) = .............. 2- #H,/ BJE) 3 AJ 'D-'D'* 'D*'DJ) ( ( 3 ) = 48 ........................................................................................................... ............................................................................................................ ............................................................................................................ 3- #+(* #F 'D*7(JB *('JF . ............................................................................................................ ............................................................................................................ ............................................................................................................ ............................................................................................................ E+'D (4) : ( : ( ! ( -J+ ( ( 3 ) = 32 + 2 . ,/ : (1) ( ( 4 ) = ........................... (2) ( ( - 4 ) = .................... (2) GD ( *('JF  DE'0'  ............................................................................................................ ............................................................................................................ ............................................................................................................ E+'D (5) : ( : ( ! ( -J+ ( ( 3 ) = 3 - 5 . (1) #H 5H1) 'D9F'51 3  2  1  0  -1  -2  -3 . (2) #H,/ BJE) F AJ 'D-'D'* 'D*'DJ) ( ( F ) = 0  ( ( F ) = 42 ( ( F ) = -11 (3)#/13 FH9 'D*7(JB %F C'F *('JF' #H 4'ED' #H *B'(D' .  [3] *-5JD 'D*7(JB'* : Composition  ( 1  ( 2 *7(JB'F 9DI - CE' GH EH6- AJ 'D13E 'D3GEJ  ( 1 ( 2  ( 2 ( ( 1 (( ) = .............................................. ( 2 ( ( 1 (( ) = .............................................. ( 2 ( ( 1 ( ) = .............................................. ( 2 ( ( 1 (/ ) = .............................................. ( 2 ( (1 *91JA : 'D*7(JB 'DE-5D D*7(JBJF ( 1 : ( ( (  ( 2 : ( ( ( . GH *7(JB ( 2 ( ( 1 : ( ( ( . 'DE91A ('DB'9/G ( 2 ( ( 1 ( 3 ) = ( 2 (( 1 ( 3 ) ) DCD 3 ( ( E+'D (6) : ( : ( ! ( *7(JB E91A ('DB'9/) ( (3) = 3 + 2 ( : ( ! ( *7(JB E91A ('DB'9/) ( (5) = 52 + 1 ',( 9E' JDJ : ( ( ( ( 3 ) = ........................................................................................... (2) ( ( ( (3 ) = ............................................................................................ (3) ( ( ( (3 ) = ............................................................................................ (4) ( ( ( (- 6 ) = ......................................................................................... (5) ( ( ( (0) = .............................................................................................. (6) ( ( ( ( - 15 ) = ....................................................................................... GD 9EDJ) *-5JD 'D*7(JB'* %(/'DJ)  .............................................................................  [4] E9CH3 'D*7(JB :  (1) *7(JB *('JF *7(JB 4'ED (3) *7(JB *B'(D *91JA : 1 - %0' C'F 'D*7(JB * *('JF A%F E9CH3) DJ3 *7(JB . 2- %0' C'F 'D*7(JB * 4'ED A%F E9CH3) DJ3 *7(JB . 3- %0' C'F 'D*7(JB * *B'(D ( *('JF H4'ED E9' ) A%F E9CH3) *7(JB HJ1E2 DG ('D1E2 * -1. E+'D (7) : * : ( ( ( . -J+ * ( F ) = F  2 DCD F ( ( . 1- '/13 FH9 'D*7(JB . 2- #H,/ *-1 %F C'F 0DC EECF' . 3- #H,/ *-1 ( 9 )  *-1 ( - 9 ) . E+'D (8) : ( : ( ( ( . -J+ ( ( 3 ) = 32 - 1 . ( : ( ( ( . -J+ ( ( 3 ) = 3 + 1 . #H,/ 'D5H1) 'D9C3J) DD9//JF 3  - 3 ('D*7(JB ( . %F #ECF . GD 'D*7(JB ( E*('JF  DE'0'  GD DG E9CH3  DE'0'  GD JH,/ DD*7(JB ( E9CH3  #H,/ 'DB'9/) 'D9C3J) DG . #H,/ ( ( ( -1 ( 8 ) = ........................................................................................ ( -1 ( ( ( 4 ) = .......................................................................................... 'D#3*F*', :.............................................................................................................  *E@@@@@@@@@@'1JF 9DI 'D*7(JB 31 : (  ( *7(JB'F EF ( ( ( -J+ ( ( 3 ) = 3 + 5  ( ( 3 ) = 32 - 1 1- #H,/ ( ( 3 )  ( ( -7 )  ( 5 ( ( 2 )  ( 5 ( ( 2 ) #H,/ 'D5H1) 'D9C3J) DD9// 8 ('D*7(JB ( . GD DG E9CH3  DE'0'  'C*( 'DB'9/) 'D9C3J) DD*7(JB (. +E #H,/ (-1 ( 7 )  (-1 ( 5 ) . 32 : (  ( *7(JB'F EF ( ( ( -J+ (( F ) = 3 F + 2  ( ( F ) = F2 . 1- #H,/ ( 5 ( ( -3 )  ( 5 ( ( 3 ) 2- #H,/ BJE F -J+ ( ( F ) = 16  ( ( F ) = 9  ( (F) = - 16 3- '+(* #F ( *('JF H4'ED 4- '+(* #F ( :J1 *('JF . 5- 'C*( B'9/) (-1 . #H,/ (-1 ( -13 ) . 33 : ( : ( ( ( -J+ ( ( 3 ) = 3 3  2  ( : ( ( ( -J+ ( ( 3 ) = 32 + 4 1-'13E E.77 3GEJ :J1 C'ED DD*7(JB ( EH6-' 5H1) 'D#9/'/ 3  2  1  0  -1  -2 2-#H,/ (5 ( ( -3 )  ( 5 ( ( -3 ) 3-G@D ( E*('JF  DE'0'  4- #H,/ 'D5H1) 'D9C3J) DD9// 25 ('D*7(JB (. 34 : ( : ( ( ( -J+ ( ( 5 ) = 2 5 + 1  ( : ( ( ( -J+ ( ( 3 ) = 32 - 3 1- #H,/ ( 5 ( ( 9 )  ( 5 ( ( 9 )  ( 5 ( ( 6 )  ( 5 ( ( 0 ) 2- #H,/ (-1 5 ( ( 15 )  ( 5 (-1 ( - 3 ) . '.*(@@@@@@@@'1 0'*@J (2) [1] '.*1 'D#,'() 'D5-J-) 31 : %0' C'F : (1 : ( ( ( E91A'K ('DB'9/) (1 ( 3 ) = 23 (2 : ( ( ( E91A'K ('DB'9/) (2 ( 3 ) = 3  2 A%F B'9/) (1( (2 GJ : ( ) 23  4 () 3  1 ,@) 23  2 / ) 3  2 32 : %0' C'F (: ( ( ( (-J+ ((3) = 3 + 23 A%F 'D5H1) 'D9C3J) DD9F51 3 GJ :  ( ) ( () { } ,@) { 3} / ) {0} 33 : %0' C'F ( : ( ( ( E91A@K' ('DB'9/) (( 3) = 23 A%F E/I 'D*7(JB GH : ( ) {3 : 3 9// 5-J- 2H,J} () {3 : 3 9// 5-J- A1/J} ,@) {3 : 3 9// 5-J- 2H,J EH,(} / ) {3 : 3 9// D' JB(D 'DB3E) 9DI5} [2] * : ( ( ( . -J+ * ( F ) = F3  1 . (1) #H,/ * ( 5 ) = .............................................................................................. #H,/ 'D5H1) 'D9C3J) DD'9/'/ - 2  7 . (3) (JF FH9 'D*7(JB E9 'D*9DJD . (4) #H,/ * ( * ( 2 ) = (5) * ( *-1 ( 5 ) =  'D('( 'D+'D+ 'DGF/3) 'DE3*HJ) Aplan Geometry 'DE6@@D9@'* Polygons  [1] *91JA 'DE6D9 :  'DE6D9 GH '*-'/ +D'+ B79 E3*BJE) #H #C+1 (-J+ : 'DB79 'DE3*BJE) **B'79 9F/ #71'AG' AB7 . CD 71A JF*EJ %DI B79*JF E3*BJE*JF AB7 . D' *H,/ B79*'F E3*BJE*'F *4*1C'F AJ 71A H'-/ 9DI %3*B'E) H'-/) . [2] 'FH'9 'DE6D9'* : J3EJ 'DE6D9 (9// 'D1$H3 ( (9// 'DB79 'DE3*BJE) 'DECHF) DG ) HJFB3E 'DI FH9JF : E6D9 E-/( Convex Polygons: %0' HB9 (C'EDG AJ ,G) H'-/) ('DF3() DCD E3*BJE J-HJ 6D9' EF #6D'9) E6D9 EB91 Concave Polygons: %0' DE JB9 (C'EDG AJ ,G) H'-/) ('DF3() DCD E3*BJE J-HJ 6D9' EF #6D'9) .  [3] E-J7 'DE6D9 : Perimeter of Polygon 7HD E-J7 'DE6D9 = E,EH9 #7H'D 'DB79 'DE3*BJE) 'DECHF) DG .  [4] 'DE6D9'* 'DE*4'(G) :Similar Polygons FBHD 9F E6D9JF DGE' 'D9// FA3G EF 'D#6D'9 #FGE' E*4'(G'F %0' : 'D2H'J' 'DE*F'81) E*3'HJ) . 'D#6D'9 'DE*F'81) E*F'3() . ...................................................................... ..........................................................................  *0C@@@@@@@@@1 'DE6D9'F 'DE*7'(B'F E*4'(G'F . 'DE6D9'F 'DE4'(G'F D+'D+ E*4'(G'F . 'DE6D9 'DE7'(B D#-/ E6D9JF E*4'(GJF J4'(G 'D#.1 . [5] F8@@@1JG :  %0' *4'(G) E6D9'F A%F : 'DF3() (JF 7HDI 6D9JF E*F'81JF = F3() 'D*4'(G = + . 'DF3() (JF 7HDI E-J7I E6D9JF E*4'(G'F = F3() 'D*4'(G = + . ..................................................................... ..................................................................... ..................................................................... ...................................................................... E+'D (1): E6D9'F E*4'(G'F 7HD' 6D9JF E*F'81JF GE' 3 3E  5 3E . A%0' C'F E-J7 'D#5:1 J3'HJ 60 3E . A#H,/ E-J7 'D#C(1 . [ 1003E ] ............................................................................................................................... ............................................................................................................................... ............................................................................................................................... ...............................................................................................................................   [6] *4'(G 'DE+D+'* : Similar Triangles 3(B #F *91A* 9DI *4'(G 'DE+D+'* HJ*E 0DC 9F/E' : J*4'(G E+D+'F %0' *3'H* 2H'J' #-/GE' E9 2H'J' 'D#.1 'DEF'81) DG' . J*4'(G E+D+'F %0' *3'H* 2'HJ) EF #-/GE' E9 2'HJ) EF 'D#.1 H*F'3( 6D9' #-/I 'D2'HJ*JF E9 6D9' 'D#.1I . J*4'(G E+D+'F %0' *F'3(* #6D'9GE' 'DE*F'81) . */1J( : '13E E+D+'* *H6- CD -'D) .  [7] F8@@@1J) :  1- %0' *4'(G) E+D+'F A%F 'DF3() (JF 7HDJ '1*A'9JF E*F'81JF AJGE' = F3() 'D*4'(G = + . ............................................................................................................................ 2-%0' *4'(G) E+D+'F A%F 'DF3() (JF 7HDJ EF5AJ 2'HJ*JF /'.DJ*JF E*F'81*JF AJGE' = F3() 'D*4'(G = + . ......................................................................................................................... %0' *4'(G) E+D+'F A%F 'DF3() (JF 7HDJ E*H37JF E*F'81*JF AJGE' = F3() 'D*4'(G = + .   ...................................................................................................................... %0' *4'(G) E+D+'F A%F 'DF3() (JF E-J7JGE' = F3() 'D*4'(G = + . ( ............................................ %0' *4'(G) E+D+'F A%F 'DF3() (JF E3'-*JGE' = E1(9 F3() 'D*4'(G = +2 (...................................... E+'D (2) : E+D+'F E*4'(G'F E-J7JGE' 28 3E  42 3E . A%0' C'F '1*A'9 'DE+D+ 'D#C(1 3 3E . A#H,/ '1*A'9 'D#5:1 'DEF'81 DG. [ 2 3E ] E+'D (3) : E+D+'F E*4'(G'F E3'-*JGE' 9 3E2  16 3E2 . A%0' C'F '1*A'9 'D#5:1 6 3E , AE' 7HD 'D#1*A'9 'DEF'81 DG AJ 'D#C(1 . [ 8 3E ]  [8] F8@@@@1J@) :  1- %0' *4'(G) E6D9'F A%FG JECF *B3JE CD EFGE' %DI E+D+'* **4'(G E9 F8'&1G' AJ 'DE6D9 'D#.1 . 2- %0' *4'(G) E6D9'F A%F 'DF3() (JF E3'-*JGE' = E1(9 F3() 'D*4'(G = +2 . E+'D (4) : E6D9'F E*4'(G'F E-J7JGE' 150 3E  25 3E . A%0' C'F* E3'-) 'DE6D9 'D#5:1 18 3E2 . AE' E3'-) 'D#C(1 . [ 108 3E2 ] E+'D (5) : 3 5 9  3N 5N 9N E+D+'F E*4'(G'F AJGE' 7HD [ 3 5 ] = 4 3E  7HD [ 5 9 ] = 5 3E  7HD [ 3 9 ] = 7 3E  E-J7 3N 5N 9N J3'HJ 24 3E . #H,/ '7H'D '6D'9 'DE+D+ 3N 5N 9N . [ 6 3E  7.53E  10.5 3E ] E+'D (6) : ( ( E+D+ E3'-*G 72 3E2  %0' C'F / // ( . (  / = 8 3E  ( = 12 3E  #-3( E3'-) 'DE+D+ ( / . [ 32 3E2 ]  /  ( [9] F8@@@@@1J@) :  1- E,EH9 'D2H'J' 'D/'.DJ) DE6D9 9// #6D'9) F 6D9 = ( F  2 ) 180 R . 2- 9// 'D#B7'1 'DE13HE) EF #-/ 1$H3 E6D9 9// #6D'9) F 6D9 = ( F  3 ) . 3- 9// 'DE+D+'* 'DF'*,) EF 13E 'D#B7'1 EF #-/ 'D1$H3 = ( F  2 ) .  4- 9// #6D'9 E6D9 E,EH9 2H'J'G 'D/'.DJ) , R = + 2 . E+'D (7) : E6D9 0H 24 6D9 . #H,/ (1) E,EH9 2H'J'G . (2) 9// 'D#B7'1 . (3) 9// 'DE+D+'* . [ 3960 R  21 B71  22 E+D+ ] E+'D (8) : E6D9 E,EH9 2H'J'G GJ 2340 R . #H,/ 9// #B7'1G . [ 13 B71 ]  'DE6D9'* 'DEF*8E)Regular Polygons [1] *91JA : 'DE6D9 'DEF*8E *CHF #6D'9G E*3'HJ) H2H'J'G E*3'HJ) . E-J7 'DE6D9 'DEF*8E = 9// 'D#6D'9 7HD 'D6D9 .  BJ'3 'D2'HJ) 'DH'-/) DE6D9 FHFJ EF*8E= 3 =  9// #6D'9 E6D9 EF*8E BJ'3 #-/I 2H'J'G 3 R = 'DE6D9 'DEF*8E *E1 (1$H3G E-J7 /'&1) H'-/G . E+'D (9) : E6D9 EF*8E 9// #6D'9) 12 6D9 . H7HD 6D9) 7 3E . #H,/ 7HD E-J7G HBJ'3 #-/I 2H'J'G. [ 84 3E  150 R ] E+'D (10) : E6D9 EF*8E BJ'3 #-/I 2H'J'G 140 R . #H,/ 9// #6D'9) . [ 9 ]  [2] 'DE6D9'* 'DEF*8E) AJ 'D/'&1) :  1- *3EJ 'D/'&1) 'D*J *E1 (1$H3 'DE6D9 'DEF*8E ('D/'&1) 'D.'1,J) DDE6D9 . 2- *3EJ 'D/'&1) 'D*J *E3 #6D'9 'DE6D9 'DEF*8E EF 'D/'.D ('D/'&1) 'D/'.DJ) DDE6D9 . */1J( : '13E E6D9 EF*8E 9// '6D'9G 5  8 /'.D /'&1) . ( E3*./E' #/H'* 'DGF/3) ) .  [3] E3'-) 'DE6D9 'DEF*8E :The area of Regular Polygons E3'-) 'DE6D9 'DEF*8E = F5A -'5D 61( 7HD E-J7G 7HD 9'E/G .  E 'D9'E/ [4] (96 'DE6D9'* 'DEF*8E) /'.D /'&1) : 'DE6D97HD 'D6D9 = D7HD 'D9'E/'DE3'-)'DE+D+ 'DEF*8ED = FB [3 / ( FB( D2 [3 /'DE1(9D = FB [2 /( FB [2 /D2 = 2 FB2'D3/'3J 'DEF*8ED = FB [2 /  (FB [3 /( 3 [3 / D2 E+'D (11) : #H,/ E3'-) E+D+ EF*8E 7HD 6D9) 12 3E .  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(2)E9'/D) 'D/1,) 'D+'FJ) : Quadratic eqation............................................................. */1J( : 9JF 'D/1,) H'DE9'ED'* DCD EF 'DE9'/D'* 'D*'DJ) : 2 34  33 + 5 32  9 = 0 ........................................................................................................ ........................................................................................................ 37  1 = 0 ........................................................................................................ ........................................................................................................ 3 32  8 3 + 5 = 0 ........................................................................................................ ........................................................................................................ 2 3 + 6 = 32  5 3 ........................................................................................................ ........................................................................................................ ........................................................................................................ [2] E9'/D) 'D/1,) 'D+'FJ) : Quadratic eqation */1J( : -// BJE 'DE9'ED'* (  (  DCD EF 'DE9'/D'* (9/ #F *69G' AJ 'D5H1) 'DBJ'3J) . 6 32  5 3 + 1 = . ........................................................................................................ ........................................................................................................ 3 3  8 = 2 32 ........................................................................................................ ........................................................................................................ [3.- /+/ /5/ = 3 3 ........................................................................................................ ........................................................................................................ ........................................................................................................ [3] -D E9'/D) 'D/1,) 'D+'FJ) :  'DEEJ2 = 2 = (2  4 ( (   E+'D (1): 'H,/ 'DEEJ2 +E (JF FH9 'D,01JF DCD EF 'DE9'/D'* 'D*'DJ) :  (1) 4 32 + 25 = 20 3 . (2) 3 32 + 3 + 1 = 0 (3) 32 = 5 3  6 . E+'D (2) : -D 'DE9'/D'* 'D*'DJ) : (1) 3 32  3  3 = 0 (2) 4 32  6 3 + 1 = 0 (3) 2 32  5 3 = - 3 (4) 3 = [3.- /+/ /3/ .  [4] 'D9D'B) (JF 'DE9'ED'* H'D,0H1 :  %0' C'F D1  D2 ,01'F DDE9'/D) ( 32 + ( 3 + = . -J+ ( `" 5A1 .  A%F ( D1 + D2 =  D1 0 D2 = ( 3  D1 ) ( 3  D2 ) = 0 ( 32  ( D1 + D2 ) 3 + D1 D2 = 0 E+'D (3) : #H,/ E,EH9 H-'5D 61( ,01J 'DE9'/D) 3 32  5 3 + 2 = 0 E+'D (4) : 'H,/ 'DE9'/D) EF 'D/1,) 'D+'FJ) 'D*J ,01'G' - 3  7 .  [5] 'DE9'/D'* 'D,(1J) AJ E*:J1JF : System of Linear equations  1- E9'/D) 'D/1,) 'D#HDI : ( 3 + ( 5 + = 0 2- E9'/D) 'D/1,) 'D+'FJ) : ( 32 + ( 52 + 3 5 + / 3 +  5 + H = 0 E+'D (5): E+D CD EF 'DE9'/D'* 'D*'DJ) AJ 'DE3*HI . 2 3 + 5 = 1 . 5 = 32 + 1  32 + 52 = 4 'DGF/3) 'D*-DJDJ) [1] E9'/D) 'D.7 'DE3*BJE : Linear equation  'D5H) 'D9'E) DE9'/D) 'D.7 'DE3*BJE GJ ( 3 + ( 5 + = 0 H*91A ('DE9'/D) 'D.7J) */1J( : E+D 'DE9'/D) 2 3 + 3 5 = 6 AJ 'DE3*HI . 35 (1) E9'/D) 'DE3*BJE 'DEH'2J DE-H1 5 69 ( = 0 AJ 'DE9'/D) ( 3 + ( 5 + = 0 ( 3 = .............. E9'/D) E3*BJE // E-H1 'D5'/'* . */1J( : '13E 'DE3*BJE 3  3 = 0 (2)E9'/D) 'DE3*BJE 'DEH'2J DE-H1 3 69 ( = 0 AJ 'DE9'/D) ( 3 + ( 5 + = 0 ( 5 = .............. E9/D) E3*BJE // E-H1 'D3JF'* . */1J( : '13E 'DE3*BJE 5 + 2 = 0 [2] *91JA : 1- E9'/D) 'DE3*BJE (/D'D) 'DEJD H'D,2! 'DEB7H9 EF E-H1 5 .  5 = - 3 - ( 5 = ( 3 + /  ( ( 'DEJD Slope) = -  / ( 'D,2! 'DEB7H9 EF E-H1 5 ) = - */1J( : 69 'DE9'/D'* 'D.7J) 'D*'DJ) 9DI 'D5H1) 5 = ( 3 + / +E 'H,/ 'DEJD H'D,2! 'DEB7H9 EF E-H1 5 . (1) 3 3 + 2 5 - 5 = 0 (2) 3  3 5 = 6 */1J( : 'C*( E9'/D) 'DE3*BJE 'D0J EJD) 5 HJB79 E-H1 5 AJ 'DFB7) ( 0  -3 ) .  [3] *91JA : ( ( 31  51 )  ( ( 32  52 )  F ( 3  5 ) FB'7 AJ 'DE3*HI 'D/JC'1*J .  EJD 'DE3*BJE ( ( = ( =  E9'/D) 'DE3*BJE (/D'D) 'DEJD ( HFB7) ( GJ = ( ( ( 5  51 ) = (( 3  31 ) .  E9'/D) 'DE3*BJE (/D'D) 'DFB7*JF (  ( GJ =  EJD 'DE3*BJE ( 3 + ( 5 + = 0 GH ( = - E+'D (6) : ( ( 3  -2 )  / ( 1  4 ) FB7*'F AJ 'DE3*HI #H,/ : EJD 'DE3*BJE ( / = E9'/D) 'DE3*BJE 'DE'1 ('DFB7) ( HEJD) 5 = E9'/D) 'DE3*BJE ( / =  E+'D (7) : #H,/ EJD CD EF 'DE3*BJE'* 'D*'DJ) : 2 3 + 3 5 _ 5 = 0 3 = 2 5  1 3  5 = 0 53 + 7 = 0  [3] F81J) :  D1 : 5 = (1 3 + /1  D2 : 5 = (2 3 + /2 E3*BJE'F AJ 'DE3*HI . (1) D1 // D2 ( (1 = (2 . (2) D1 ( D2 ( (1 (2 = - 1 #H (2 = E+'D (8) : #H,/ E9'/D) 'DE3*BJE AJ 'D-'D'* 'D*'DJ) : EJD) (  2 ) HJE1 ('DFB7) ( - 3  1 ) . JE1 ('DFB7) ( 2   2 ) HJH'2J 'DE3*BJE 5 3 = 4 5  2 . JE1 ('DFB7) ( -1  5 ) H9EH/J 9DI 'DE3*BJE 2 5 = 7 3 + 1 .  *7(JB : #H,/ BJE) C AJ 'D-'D'* 'D*'DJ) : EJD 'DE3*BJE 2 5  C 3 + 1 = 0 GH ( - 1.2 ) . [ C = - 2.4 ] 'DE3*BJE'F 2 3 + 5 5 = 1  3 3 = C 5 + 6 E*H'2J'F . [ C = - 7.5 ] 'DE3*BJE'F 2 3 + 5 5 = 1  3 3 = C 5 + 6 E*9'E/'F . [ C = 2.4 ] 'DE3*BJE C 3 + 2 5 - C  1 = 0 JE1 ('DFB7) ( 3  5 ) . [ C = - 4.5 ] F8'E E9'/D*JF EF 'D/1,) 'D#HDI AJ E*:J1JF System of Linear equations */1J( : -D 'DF8'E 3 + 5 = 5  3  5 = 1 . { ( 3,2 ) } [1] 'DE-//) : Determinant The  ( = = ( / - ( . E+'D (10) : #H,/ BJE) :CD EF  = ...................................................................................  = ...................................................................................  [2] -D F8'E E9'/D*JF ('DE-//'* : (1 3 + (1 5 = 1  (2 3 + (2 5 = 2  ( =  ( 3 =  ( 5 = 3 = ............................  5 = ............................... E,EH9) 'D-D { ( 3  5 ) } . E+'D (11) : #H,/ -D 'DF8'E 2 3 + 3 5 = 4  2 5  3 = 5 . { ( - 1  2 ) } [3] F8@@@@1J@) : D-D F8'E E9'/D*JF (1 3 + (1 5 = 1  (2 3 + (2 5 = 2  (1) ( `" 5A1 ( 'DF8'E DG -D H-J/ GH 3 =  5 = (2) ( = ( 3 = ( 5 = 5A1 ( 'DF8'E DG 9// :J1 EF*G EF 'D-DHD . ( :J1 E-// 'D-D ) . HJA31 GF/3J' #F 'DE3*BJE'F E*7'(B'F . (3) ( = 5A1  ( 3 `" 5A1  ( 5 `" 5A1 ( 'DF8'E DJ3 DG -D  E,EH9) 'D-D = ( . HJA31 GF/3J' #F 'DE3*BJE'F E*H'2J'F . E+'D (12) : ('3*./'E 'DE-//'* -D 'D#F8E) 'D*'DJ) : 2 3 = 5 + 3  4 3  5 5  6 = 0 { ( 1.5  0 ) } 2 3  3 5 = 5  3 + 2 5  4 = 0 { ( - 2  3 ) } 3 3  2 5 = 0  3 + 5 = 5 { ( 3,2 ) } E+'D (13) : #H,/ BJE) C 'D*J *,9D 'DF8'E DJ3 DG -D' : (1) C 3 + 4 5 = 0  2 3 _ 6 5 = 7 . [ C = - 8] (2) C 3 + 4 5 = 3  9 3 + C 5 = 1 . [ C = DI 6 ]  [4] (9/ FB7) 9F E3*BJE :  (1) (9/ 'DFB7) F ( 31  51 )  'DE3*BJE ( 3 + ( 5 + = 0 GH   (2) (9/ 'DFB7) H ( 0  0 ) 9F 'DE3*BJE ( 3 + ( 5 + = 0 GH . E+'D (14) : #H,/ (9/ 'DFB7) 'DE97'G 9F 'DE3*BJE 'DE97I : 'DFB7) F = ( (3  - 1 )  'DE3*BJE 4 3  3 5 + 5 = 0 [ 4 H-/'* ]  'DFB7 H ( 0  0 ) H'DE3*BJE 2 3  5 = 7 . [ ]  F8'E E9'/D*JF EF 'D/1,) 'D+'FJ) AJ E*:J1JF E+'D (15) : #H,/ -D 'DF8'E 3  5 = - 1  4 32 + 52 = 25 . { ( 2  3 )  ( - 2.4  - 1.4 ) } E+'D (16) : #H,/ -D 'DF8'E 2 32  3 52 = 5  3 32 + 4 52 = 16 . { ( 2  1 )  ( 2  -1 ) ( - 2  1 )  ( - 2  - 1 ) } E+'D (17) :#H,/ #(9'/ 'DE3*7JD 'D0J E-J7G 14 3E H7HD B71G 5 3E . [ 33E  4 3E ]  F4@@@@'7 -'HD -D 'D#F8E) 'D*'DJ) 9BDJ'K . (1) 3 3  4 5 = 25  32 + 52 = 25 { ( 3  4 )  ( 3  - 4 ) } (2) 3 + 5 = 3  32 + 52 = 9 { ( 3  0 )  ( 0  3 ) } (3) 5 = 3  5 = 32 { ( 0  0 )  ( 1  1 ) } (4) 3  5 = 1  32  5 = 1 { ( 0  - 1 )  ( 1  0 ) } 'D/'&@@@@@@@@1) The circle [1] 'D5H1 'DBJ'3J) DE9'/D) 'D/'&1) :Equation of the circle  FB FB F ( (  ( ) H (1) 32 + 52 = FB2 (2) ( 3 - ( )2 + ( 5  ( )2 = FB2 E+'D (18) : 'C*( E9'/D) CD EF 'D/H'&1 'D*'DJ) : ((  4 3E ) ( ................................................................................ ((  ( 3E ) ( ................................................................................ F ( -1  3 )  FB = 5 3E ( ................................................................. F( (  - ( )  FB = 2.5( .................................................................. E+'D (19) : 9JF '-/'+J'* 'DE1C2 H7HD F5A B71 CD EF 'D/H'&1 'D*'DJ) : 32 + 52 = 18 ( ......................................................................... ( 3 + 2 )2 + ( 5  1 )2 = 7 ( .......................................................  [2] 'D5H1) 'D9'E) DE9'/D) 'D/'&1) :  'DE9'/D) 32 + 52 + 3 + / 5 + = 0  (1) %0' C'F* ( )2 + ( )2 - < 5A1 . ( D' *E+D /'&1) -BJBJ) .  (2) %0' C'F* ( )2 + ( )2 - = 5A1 ( *E+D FB7) GJ (  ) .  (3) %0' C'F* ( )2 + ( )2 - >5A1 . ( *E+D /'&1) -BJBJ) .  E1C2G' (  )  FB = ( )2 + ( )2 - .  *0C@@@@@@@@@@1 1- E9'ED 32 = E9'ED 52 . 2- E9'ED 3 5 = 5A1 . E+'D (20): (JF #J 'DE9'/D'* 'D*'DJ) *E+D /'&1) E9 0C1 'D3(( : +E 9JF 'DE1C2 H7HD F5A 'DB71 AJ -'D) 'D/'&1G 32  5 2 + 4 3  6 5 + 2 = 0 2 32 + 2 52 - 10 3 + 4 3 5 + 4 5 - 8 = 0 32 + 52  6 3 + 10 5 + 50= 0 332 + 3 52 + 12 3 - 18 5 + 15 = 0 232 + 2 52  8 3 + 12 5  12 = 0  [3] #H6'9 FB7) ('DF3() DD/'&1) :  F ( 31  51 )  E = E1C2 'D/'&1) F5A B71G' FB (   F  F E F EE'3 DD/'&1) E+'D (21) : -// EHB9 'DFB7) F ( 1 2 )('DF3() DD/'&1) ( 3 + 3 )2 + ( 5  4 )2 = 20 . E+'D (22) : #H,/ E9'/D) 'D/'&1) 'D*J E1C2G' ( 0 2 ) H*E1 ('DFB7) ( 0  7 ) . E+'D (23) : -// EHB9 'DFB7) F (2  4 ) ('DF3() DD/'&1) 32 + 52  63  14 = 0 E+'D (24) :,/ E9'/D) 'DEE'3 DD/'&1) 'D*J E1C2G' F ( - 2  1 ) 9F/ 'DFB7) / ( 3  5 ) . E+'D(25) :#H,/ %-/'+J E1C2 H7HD F5A B71 'D/'&1) 32 + 52  2 3  4 5  4 = 0  [4] 9D'B) 'DE3*BJE ('D/'&1)    'DE3*BJE .'1, 'D/'&1) 'DE3*BJE JB79 'D/'&1) 'DE3*BJE EE'3 DD/'&1) D' JB79 'D/'&1) JB79 'D/'&1) AJ FB7*JF JD*BJ E9 'D/'&1) AJ F*B7) H'-/) (9/ 'D3*BJE 9F 'DE1C2 ( FB (9/ 'D3*BJE 9F 'DE1C2 ( FB (9/ 'D3*BJE 9F 'DE1C2 = FB E+'D (26) : #H,/ FB7 *B'79 'DE3*BJE 3  5 + 2 = 0 E9 'D/'&1) ( 1  0 ) HF5A B71G' 3 3E . { ( 2  4 )  ( 1  3 ) }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e" )" H"  " `" " d" ( 'D#3E / .............................. 'DA5D / ............... (( ( 5..5 'DC*'( 5 10 'DC*'( 5 13 'DC*'( 5 13 (( (  ((( (( ( ( (  'DC*'( 5 21 'DC*'( 5 21 'DC*'( 5 25 'DC*'( 5 28 'DC*'( 5 30 (~ ~ (~  ( (~ ( ( ~  ( ()555..5..  ( 3 ( 5 7 9 ( 3 ( 7 ( 3 ( 5 7 ( 2 ( 0 1 / 3 0 D 1 E 2 F 3 H ( D ( E F / H 'DC*'( 5 39 'DC*'( 5 42 'DC*'( 5 44 'DC*'( 5 53 'DC*'( 5 54 'DC*'( 5 63 'DC*'( 5 70 'DC*'( 5 80 3 2 'DC*'( 5 92 'DC*'( 5 95 'DC*'( 5 98 'DC*'( 5 100 'DC*'( 5 106 'DC*'( 5 107 E,EH9 'D2H'J' 180 ( F  2 ) 180 F 360 180  3 'DC*'( 5 114 'DC*'( 5 116 'DC*'( 5 121 'DC*'( 5 128 3 R / 180 7 / 7 3 180 3 7 4 5 7 6 'DC*'( 5 130 / 2 7 'DC*'( 5 142 2 < 5A1 & . DJ3 DDE9'/D) -D AJ ( . 2 = 5A1 .... DDE9'/D) ,01'F E*3'HJ'F GE' - 2 > 5A1 .. DDE9'/D) ,01'F -BJBJ'F E.*DA'F GE' 3 = ( 2 ( ( ( - ( (2  4 ( 2 ( 'DC*'( 5 147 'DC*'( 5 154 - ( ( ( 'DC*'( 5 159    52  51 32  31 5  51 3  31 5  51 3  31 52  51 32  31 ( ( -1 E1 'DC*'( 5 166 'DC*'( 5 168 'DC*'( 5 179 ( / 2 5 7 1 6 2 (1 (1 (2 (2 'DC*'( 5 182 'DC*'( 5 187 (3 ( (5 ( ( 31 + ( 51 + (2 + (2 (2 + (2 'DC*'( 5 193 'DC*'( 5 199 'DC*'( 5 209 'DC*'( 5 210 2 2 2 - 2 *E'1JF ( 5 219 ) 8 & 16 & 18 & 19 2 / 2 / 2 - / 2 / 2 - / 2 / 2 F E < FB ( F *B9 /'.D 'D/'&1) . F E = FB ( F *B9 9DI 'D/'&1) . F E > FB ( F *B9 .'1, 'D/'&1) . 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