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   @-HH,, ,,..HH>>HH,,,,  ////>>HG ****   "**//HGHH,,HG 9HG**..+ HG -       21HH,,>> H I J?2$E2;VVqeh3r$$E2fY_,I!)ir$|ka@O+8ki 0AA fދ3ff33@89:g4kdkd@z[ 0vppp@  g4+d+d@z[ 0rp@ ppʚ;ʚ;<4ddddl|- 0X80___PPT10 pp?  %O == ;@ Uvjeti koriatenja prezentacije:!!( Autorica: Sonja Bani, prof. Ovu prezentaciju dozvoljeno je koristiti za rad s u enicima. U tu svrhu dozvoljeno ju je i izmijeniti prema svojim potrebama. Nije dozvoljeno koristiti i prikazivati ovu prezentaciju na predavanjima, u lancima, knjigama ili objavljivati na CD-u ili internetu. Za te i sli ne namjene potrebno je tra~iti i dobiti pristanak autorice.6sZR  JSKUPOVIGfSkupovi i neka njihova svojstva Sonja Bani, prof.0!!*Ova prezentacija pomoi e vam da steknete neka osnovna znanja o skupovima.LLPodatke koji vam se ine va~ni mo~ete, ako ~elite, zapisati u bilje~nicu. Ako ~elite ii dalje, kliknite na lijevu tipku miaa.  " # % & )/03567/T!$4  ` f̙f` \'݉f̙ft` 3v;̙3fff` Q%]m6fft` JU;f̙ft` fffff` f33f̙f3` fff̙` f33f̙f3>?" dd@,?pKd@n pA@d`hP n?" dd@   @@``PT    @ ` `:p>> PH (    6 " `}  T Click to edit Master title style! !$  0H " `  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0ܲ "` `  ^*    0t "`   \*   0 "` `  \*   6|" D0 B  s *DjJ" ,$ 0   6"@  D0    6"@  D0 H  0޽h ? f33f̙f3___PPT10i. η+D=' = @B +  Level   (    6b7 #" ` 7 T Click to edit Master title style! !  0Dp7 " ` |  7 W#Click to edit Master subtitle style$ $  07 "` ` 7 ^*    0@7 "`  7 \*   07 "` ` 7 \*     ",$D  0h  s *" h   s *" h   s *" H  0޽h ? f33f̙f3___PPT10i. η+D=' = @B + 0 TL0D(  D D 0@ P    P*   D 0`F     R*  d D c $ ?   D 0lL  0  xRClick to edit Master text styles Second level Third level Fourth level Fifth level S D 6N _P   P*   D 6T _   R*  H D 0޽h ? 3380___PPT10.0/cS$  $(  r  S 7 `}  7 r  S 7 ` 7 H  0޽h ? f33f̙f380___PPT10. E= TL@(     B78c8c?<Tt 7 x  c $|7 >  7   <7*j " H  0޽h ? 33___PPT10i.">+D=' = @B +  `(  x  c $7 `}  7 x  c $7 ` 7   <`7 ,$ 0 $Zadatke rjeaavajte u bilje~nicu. Nakon ato ste rijeaili zadatke, mo~ete klikom na lijevu tipku miaa provjeriti rjeaenja.yy  <H7 _%,$  0 t@}elim vam ugodan i uspjeaan rad!!!H  0޽h ? fff  ___PPT10 .G2+rOD ' = @B D ' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*+p+0+7 ++0+7 +@" T L p((  (2  ( 0<j7,$ 0 jSkup je osnovni matemati ki pojam koji ne definiramo.66  ( <pfJM "  ( 0= ,$D  0 V esto smo spominjali skupove brojeva... skup prirodnih brojeva: N={1,2,3,4,....} skup parnih brojeva: A={2k : k je cijeli broj}6)Y ( 0% s,$D  0 ,ili skupove to aka u ravnini: Kru~nica je skup to aka ravnine jednako udaljenih od vrste to ke, srediata kru~nice.(uV2 ( 6o ;,$D  0$ ( <d ",$D 0 8Skup ine njegovi elementi. 0\ ( <`# ,$D 0 ~Element skupa je osnovni matemati ki pojam koji ne definiramo."@?H ( 0޽h ? fff___PPT10l.#P?+D`' = @B D' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*(D' =%(D+' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*(++0+( ++0+( ++0+( ++0+( +V J B  8(  8R 8 0XHI Skupove obi no ozna avamo velikim slovima A, B, C,..., a njihove elemente malim a, b, c, d,....``^ 8 6j 8 S A ??"?NN  8 0N7 B   8 0?S s  B   8 0T 7+ > R  8 <\X ,$  0 Da je a element skupa A krae zapisujemo koristei znak aA ( itamo: a je element skupa A) b:,:!  8 <b 7 ,$  0 |Da b nije element skupa A krae zapisujemo koristei znak bA ( itamo: b nije element skupa A)~a:<"H 8 0޽h ? fff  ___PPT10 ..'޺+oUD ' = @B D ' = @BA?%,( < +O%,( < +D' =%(%(D+' =%(D' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* 8%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* 8D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* 8D+' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 8%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* 8D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* 8+p+0+ 8 ++0+ 8 + yq  (  0  <dsZF Skup je zadan kada je to no odreeno koji elementi ine taj skup. "DC/l @, @,,$D   0,  <x@ , z Skup mo~emo zadati tako da nabrojimo sve njegove elemente:">=  <~# m FA={a, e, i, o, u }  <# R% I B={2,4,6,8}" bl 7   7,$D   0@  <7   Ili mo~emo odrediti koji elementi pripadaju skupu pomou njihovih svojstava:MM   <4  W#B={x | x je paran broj manji od 10}$$   <d "  W#A={samoglasnik u hrvatskoj abecedi}$$H  0޽h ? fff  ___PPT10 .X+@D ' = @B D ' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D' =4@BBB B%( D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =4@BBBB%( D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*+  &(    <' BZADATAK1: a) Zapiai navoenjem elemenata skup A viaekratnika broja 3 manjih od 30 b) Koje su od sljedeih tvrdnji istinite: 1) 3A 2) 14A 3) 2A 4) 21A (  < T ,$  0 RJE`ENJE: a) A={3, 6, 9, 12, 15, 18, 21, 24, 27} b) 1) da 2) ne 3) da 4) neNN8XH  0޽h ? fff1)___PPT10 ..@{+\QD' = @B D`' = @BA?%,( < +O%,( < +D' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' ,=+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*+8+0+ + nf ((  ( ( <Z` ?Broj elemenata nekog skupa nazivamo kardinalni broj tog skupa.R@$ff ( ( <@p fKardinalni broj skupa S ozna avamo sa |S| ili kS441) ( <pI ,$D 0 3PRIMJER: A={a, e, i, o, u }, |A| = 5 ili kA = 544- ( <A qa  B  ( <P F  D   ( < p,$D  0 b.A koliko elemenata ima skup prirodnih brojeva?//  ( <8 y ,$D 0 z$B={2,4,6,8}, |B| = 4 ili kB = 4%%  ( < Ii > H ( 0޽h ? fff3+___PPT10 .` +WD7' = @B D ' = @BA?%,( < +O%,( < +D' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* (D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* (D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* (D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* (++0+( ++0+ ( ++0+ ( += TL,(  ,b , <8d Za skupove koji imaju beskona no mnogo elemenata Ka~emo da su beskona ni skupovi. 6S> , <( 7|  6Ako je kardinalni broj skupa S prirodan broj ka~emo da je skup S kona an skup ili da ima kona no mnogo elemenata.6sA & , < 7 ,Takvi su skupovi N, Z, Q, R i C Svi oni imaju beskona no mnogo elemenata (ali ne i jednako mnogo elemenata).6nOH , 0޽h ? fff___PPT10i.c  :+D=' = @B +  0"(  0 0 <7; Skup koji ne sadr~i niti jednan element nazivamo prazan skup i ozna avamo s . O ito vrijedi: k=0Xe1 &B-   0 <]7  ZADATAK: Odredi kardinalne brojeve sljedeih skupova: 1) A={x | x je prirodan broj manji od 10} 2) B={x | x je cijeli broj manji od od 10} 3) C={x | x je prirodan broj manji od 0} 4) D={x | x je cijeli broj vei od -5 i manji od 5}8Xd 0 <0 d ,$  0 hRJE`ENJE: 1) kA=9 2) Beskona an skup 3) kC=0 4) kD=9*5$&8XH 0 0޽h ? fff ___PPT10.d-q+PD' = @B DD' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*0D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*0+8+0+0 +    !4 (  42  4 6Ԕ"`dm  > V 4 <* VDa bismo lakae stvorili predod~bu o odnosima meu skupovima koje promatramo, prikazujemo ih crte~ima. Takav na in prikazivanja skupova naziva se Venn-Eulerovi dijagrami.D, 2 4 <d ̙ffԔ" `c  > 2 4 <L$Ԕ"` 9  >   4 <('|c  AA$ 4 <t+F -  AB$ 4 </L  9u$ 4 6-   >  4 66 v  >  4 <t:P   >  4 B= / EA(  4 B@< qp  EB( !4 B@ IM 9u$H 4 0޽h ? fff___PPT10i.h+D=' = @B +o    @ (  @ @ <J7D DZa skup A ka~emo da je podskup skupa B ako je svaki element skupa A takoer i element skupa B. To ozna avamo s A B.:xs2 @ <LԔ"`7  > 2 @ BU3Ԕ"`z   >  @ <Xf  1 EB(  @ <l]f|  EA(  @ <W.  ,$   0 Na primjer, skup prirodnih brojeva je podskup skupa cijelih brojeva, N Z.\MFd @ <DhF # 92,$  0 nPrimijetimo, svaki skup je podskup samog sebe. A A:84 @ <n 7 o,$  0 XPrazan skup je podskup svakog skupa. AT-'' H @ 0޽h ? fWcJf___PPT10. p|+2D' = @B D ' = @BA?%,( < +O%,( < +D ' =%(%(D#' =%(D' =A@BBBB0B%( D' =1:Bvisible*o3>+B#style.visibility<* @%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* @D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* @D+' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*@D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*@D#' =%(XD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*@D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*@++0+ @ ++0+@ ++0+@ + L(  L  L <<  TZADACI: 1. Je li skup S={0,1,2,3,4,5,6,7,8,9} podskup skupa prirodnih brojeva?UU8XL L <S [Y  2. Neka je A={4, 6, 8, 10}. Jesu li istinite sljedee tvrdnje: a) {6} A b) {4,5,6} A c) A S d) AtJ n  L < V %,$ 0 QNe, jer 0 nije prirodan broj. L <  ,$ 0 Na) DA b) NE c) NE d) DAH L 0޽h ? fff  ___PPT10 .k+b*D ' = @B D ' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*LD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*LD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*LD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*L+p+0+L ++0+L +  PR(  P  P <   Za bilo koji skup S mo~emo promatrati skup svih njegovih podskupova. Taj skup zove se partitivni skup skupa S i ozna ava se s P (S).XV 9 CB P <8* ^Ispiai sve lanove skupa P (S) ako je S={2,3}.\0   " P <#* ,$ 0 P (S)={, {2}, {3}, {2,3}}(   P 6($ ,$  0 4U matematici esto promatramo samo podskupove nekog odreenog skupa. Za skup ije podskupove promatramo ka~emo da je univerzalni skup i ozna avamo ga s u.tu,  ,# % ?H P 0޽h ? fff  ___PPT10 ..+HD ' = @B D ' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*PD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*PD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*PD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*P+p+0+P ++0+P + 3+@8X@(  X2 !X 6<}   >  X <?/ n Unija skupova A i B je skup A B kojeg ine oni i samo oni elementi koji pripadaju ili skupu A ili skupu B ili i skupu A i skupu B.H g@ X <dG/f v :A B = {x | x A ili x B}~  X <N ] (  R A B,L -  /X# p   X 6S-  >  X 6W"   >  X <Z" b-  >   X B^ 5A  X Bb  5B  X Bfk 9u$F c  2X c  %X Bte|c  ?B"D &X BnQ 9u$2 "X B@sjJ"  N 7  > 2 #X BwjJ" ]&  >  $X BybI ?A"D^B 'X 6Do = ^B (X 6DoX= ^B )X 6Do8= v  -X 0\ r EIz definicije se jednostavno vidi da unija skupova ima ova svojstva:"FE2 .X <$ 0  1) A B= B A 2) A A B i B A B 3) A = A 4) A u = uC $$$0 XB 3X@ 0Do%  "1 XB 4X@ 0Do 3 O 8    8X3  fB X 6Do   `B 5X 0Do %  `B 6XB 0Do " `B 7XB 0Do "5 H X 0޽h ? fff___PPT10i.v)+D=' = @B + }uP\ (  \ \ <7 "   \ <ܜ  ZADATAK: Neka je: A={1,2,3,4}, B={3,4,a,b} i C={a,b,c,d} Ispiai skupove: 1) A B 2) A C 3) A B C Jesu li neki od ovih skupova jednaki? Prika~i ove skupove pomou Venn - Eulerovih dijagrama.U`>6s  l \ <H | ,$ 0 RJE`ENJE: 1) A B = {1,2,3,4,a,b} 2) A C = {1,2,3,4,a,b,c,d} 3) A B C ={ 1,2,3,4,a,b,c,d} Jednaki su skupovi A C i A B C/l2 \ 6fԔ"` ;2 \ B(Ԕ" `s   > 2  \ Bf3Ԕ"`=  >   \ Bн J2  7A  \ B"` @ (  7B  \ B<{  b  61  \ BT 62 \ B<  63 \ B  0 .  64 \ B( P N j  6a \ B P V  6b \ B 6d \ B W?  7C \ B6 92V  VA B C( ^B \ 6Do= V  \ <  ,$ 0 ^c$ f2 \ 0Ԕ"` z H \ 0޽h ? fff ___PPT10.!+D=' = @B D' = @BA?%,( < +O%,( < +D/' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*\D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*\D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-6B'blinds(horizontal)*<3<*\+p+0+\ ++0+\ + `* %(  uF -  "  {:  # 0-  >  $ 0X  >  % 64b-  >  & <D 5A ' <  5B ( <(k 9u$  <* B   <`q F$   05T; TPresjek skupova A i B je skup A B kojeg ine oni i samo oni elementi koji su i elementi skupa A i elementi skupa B. Lw V>  <0 I 6A B = {x | x A i x B}  2   < jJ?  > 2   BX"ffjJ" " %  > 2   Bh jJ"   >    B|) ?A"D   B'|c  ?B"D  B1Q 9u$^B  6Dop 3   <6 n 1  T A B.   <:0   P A B*    < 9   }GIz definicije se jednostavno vidi da presjek skupova ima ova svojstva:H Hr ! <8F1 @ sw 1) A B = B A 2) A B A i A B B 3) A = 4) A u = A(F  $03  XB @ 0Do0 A  * <U r A B00  H  0޽h ? fff___PPT10i.P+D=' = @B + !p  (    6\ ZADATAK: Neka je: A={1,2,3,4}, B={3, 4, a, b} i C={a, b, c, d} Ispiai skupove: 1) A B 2) B C 3) A Cn X*  0|f]wI,$ 0 ~RJE`ENJE: 1) A B = {3, 4} 2) B C= {a, b} 3) A C = @ = z  <xs ,$ 0 Ako skupovi A i B nemaju zajedni kih elemenata tada je njihov presjek prazan skup.S S0  <t6  ,$  0 Za skupove iji je presjek prazan skup ka~emo da Su disjunktni skupovi. Skupovi A i C iz prethodnog zadatka su disjunktni.F~ 6  4,6 2 }l   V+  V+,$D  0   0ދ"`  V+ >    xA*Wide upward diagonal"`  + >    |A.Wide downward diagonal"`F + >    <\"` + K  5A  <"`  5C  <"`(   9u$H  0޽h ? fffUM___PPT10-.@ +Y8DY' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DF' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bdiamond(in)*<3<*DM ' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(DG' =4@BBB B%( D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*++0+ ++0+ ++0+ +   :(      0'< rZADATAK: Zadani su skupovi: A={1,2,3,4,5} B={1,3,5,7} C={x | x je prirodan broj manji od 7} D={x | x je rjeaenje jednad~be x2 - 8x + 15 = 0} Ispiai i nacrtaj skupove: 1) A C 2) B D 3) A B C D 4) A B C Dx /& 8X,   6 H f,$ 0 RJE`ENJE: 1) A C={1,2,3,4,5} 2) B D={3,5} 3) A B C D={3,5} 4) A B C D={1,2,3,4,5,6,7}rg =(8X l     ,$D 0`   0    <] D  B3$    <   B5$    <L   B1$    <@P ~7  B7$    <   B2$    <(   B4$    <&  B6$ 2   B3o   D 3    <   7D 3f2   6ffo0     <= s$  7B fff2   63o Y    < ] }D  7A 3   <P`pG AC$ f2   6o `H   0޽h ? fff___PPT10u.\4+`UD' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* +8+0+  +Q& ld ,0 (  0r 0 0Z'` Razlika skupova A i B je skup A\ B kojeg ine oni i samo oni elementi koji su elementi skupa A i koji nisu elementi skupa B.(~0 o                                  * 0 <` 4A\ B = {x | x A i x B}F0       0 B   F0  8  00D  0@\ T @  00  0 00D  0 64jJ 000  J 0  Z2 0 s * `2 0 0op   0 <    OB0     0 <@ ?@  OA0     0 <vz ku20 $$  0 <$n PA\ B0   lB 0 <DjJPP  lB 0B <DjJ  f2 0 6o 8 @ tt  +0@@ @  0 <(jJ@ tt  J 0  2 0 < -o@  J 0  `2 0 0 P  0 B$0 0  OB0    0 B3 d  OA0    0 B6p t T ku20 $$ l2 0 <o P f2  0 6o@  "0 <<v* QB\A0   lB #0 <DjJ@ fB &0 6D @  fB '0B 6D P j (0 <@A  ,$D 0 .Uo imo: A\ B B \ A20 F      > )0 <GP p,$D 0 dKakvi su skupovi A\ B , B \ A i A B meusobno? n30                ,0 <|P ,$D 0 e Disjunktni! 0     H 0 0޽h ? fWcf  ___PPT10e + }LD ' = @B D\ ' = @BA?%,( < +O%,( < +D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*(0%(D' =-6B'blinds(horizontal)*<3<*(0D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*)0%(D' =-6B'blinds(horizontal)*<3<*)0D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,0%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,0D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,0++0+(0 ++0+)0 ++0+,0 +/ XP8(  8H 8 <$_n TLD___PPT9& ZADATAK: Zadani su skupovi: A={1,2,3,4,5} B={1,3,5,7} C={x | x je prirodan broj manji od 7} D={x | x je rjeaenje jednad~be x2 - 8x + 15 = 0} Nai razliku skupova: 1) A\ B 2) B \ A 3) A\ C 4) C \ A 5) (A B) \ D ~ .               8X 8 <\y7 $ 0XP___PPT92* Rjeaenja: A\ B={2,4} 2) B \ A={7} 3) A\ C= 4) C \ A ={6} 5) (A B) \ D={1} .                8X z   8  >t,$D 0Z 8 s *  8 6y] D  B3$  8 6`   B5$   8 6z   B1$   8 6lP ~7  B7$   8 6   B2$   8 6   B4$   8 6&  B6$ 2 8 <Я3o   D 3  8 6   7D 3`2 8 0ffo0   8 6= s$  7B ff`2 8 03o Y  8 6x ] }D  7A 3 8 6T`pG AC$ f2 8 6o `H 8 0޽h ? fWcfw___PPT10W+GD' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DF' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-u6Bdiamond(in)*<3<*8D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-6B'blinds(horizontal)*<3<*8+8+0+8 +     <X (  <d < 0T Komplement skupa A je skup AC kojeg ine oni i samo oni elementi univerzalnog skupa u koji nisu elementi skupa A AC= u \ A } 7$$(x   < <L"` d $,AC={x | x u i x A} (:    < <ljJ"`3   > X2 < 0jJc   < Bw{ 9u$  < BSG5  5ARB < s *D}` < < 6  \AC"  b < <|vTLD___PPT9& ~Uo imo da vrijedi: A AC= u A AC= (AC)C = A C = u uC = -$$$N%    8XH < 0޽h ? fWcf___PPT10i.p4O+D=' = @B +   @8(  @ @ <`  TLD___PPT9& ZADATAK: Neka je u ={x | x je prirodan broj manji od 50} i neka su A = {x | x je prirodan broj manji od 40} i B = {x | x je viaekratnik broja 5 manji od 50} Odredi skupove: AC A B AC \ B AC B (A B)C 4   8X2 @ <\-?$ 0LD___PPT9& >Rjeaenja: {40, 41, 42, 43, 44, 45, 46, 47, 48, 49} {5, 10, 15, 20, 25, 30, 35} {41, 42, 43, 44, 46, 47, 48, 49} {40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 5, 10, 15, 20, 25, 30, 35} 5) {41, 42, 43, 44, 46, 47, 48, 49}> A  8XH @ 0޽h ? fWcf___PPT10h.1B+'D' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DF' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =-u6Bdiamond(in)*<3<*@+8+0+@ +  0 H+(  HX H C D    H S aD 0   -podskup H H 0޽h ? 3380___PPT10.7S#  0 0T3(  TX T C D    T S hD 0   5Partitivni skup H T 0޽h ? 3380___PPT10.i 0 |4( r  4R 4 3 D    4 C nD 0   " H 4 0޽h ? 33l xp^RЀ3ÿ lHbP  LPÄ!1 4axĐ% y@( XyT0cH!=!Ff ہV30d'2b`bDD`;PtGC y'bOzC? rB x#"Z($Ftcf.HC@73(8?D4$3?OXπ( ca=c/ u3 Ѯ2 vLA^(1G8(apWBRD`ɟ ,}WFT =DҤ(T @~5r@sc .7@ %+,29EY]0_Up|_Gk) I٩/ 3Pla(CP90mh? d;Oh+'0 hp   SKUPOVI Olivetti2 Leveltisonjati33jMicrosoft PowerPointP@P'@L "@pc.qGg  I  -- @ !--'f3-- @ !--'3--%2--'f-- @ !--'3-- @ !--'@"Verdana-. 362 PUvjeti koritenja prezentacije:)!!!!!#""!!#"!."System-@"Verdana-. 3f@"Verdana--.12 fAutorica: Sonja Bani, prof.    .--.@"Verdana-. f3I2 >b,Ovu prezentaciju dozvoljeno je koristiti za        .@"Verdana-. f3hb@"Verdana--.F2 hb*rad s uenicima. U tu svrhu dozvoljeno ju   &   .--.@"Verdana-. f3C2 b(je i izmijeniti prema svojim potrebama.  %   & &&.@"Verdana-. f3I2 b,Nije dozvoljeno koristiti i prikazivati ovu          .@"Verdana-. f3b@"Verdana--.F2 b*prezentaciju na predavanjima, u lancima,   &  & .--.@"Verdana-. f362 Cbknjigama ili objavljivati na CD &    .@"Verdana-. f3 2 C-.@"Verdana-. f32 Cu ili  .@"Verdana-. f32 mb internetu .@"Verdana-. f3m@"Verdana--.:2 m". Za te i sline namjene potrebno    &  .--.@"Verdana-. f3b@"Verdana--.B2 b'je traiti i dobiti pristanak autorice.       .--.՜.+,0    )On-screen Show INA-Naftaplin uA  Arial GaramondTimes New RomanVerdana WingdingsSymbolMonotype CorsivaLevelMicrosoft Equation 3.0! Uvjeti koritenja prezentacije:SKUPOVILOva prezentacija pomoi e vam da steknete neka osnovna znanja o skupovima.Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9 Slide 10 Slide 11 Slide 12 Slide 13 Slide 14 Slide 15 Slide 16 Slide 17 Slide 18 Slide 19 Slide 20 Slide 21 Slide 22 Slide 23  Fonts UsedDesign TemplateEmbedded OLE Servers Slide Titles_esonjasonja  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:<=>?@ABCEFGHIJKMNOPQRSXRoot EntrydO)PicturesCurrent UserLSummaryInformation(;PowerPoint Document(eDocumentSummaryInformation8D