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ΠΛΗ20 ΜΑΘΗΜΑ 0.1 (ΕΚΤΥΠΩΣΗ)

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ΠΛΗ20 ΜΑΘΗΜΑ 0.1 (ΕΚΤΥΠΩΣΗ)

  1. 1. 20 0: 0.1: ! "# $ % $ & ' ( ) ' (* + ! , - ( & ) . % / , % . ! "# $% & $ ! $% ! '(%"$ & )"$ & ! "# * (-) ! "# + (-)(-) *. ,"& ! 1. $ "! $ % + : % & " $ - .& ( & : % & . %$ - $ - 1 "- 9 & % !% $ " ". ! / $ % $'"! "! $ (% %" / $% '& $% ( " . % % & },,,,,,{ ωυοιηεα=Α }9,8,7,6,5,4,3,2,1{=Β "! $ (% %" / $% '& $% ( " . % % & % $'"!& "! /! "$ " # : " % % & " - . .'. 0 & % % : " " $/ .$ , ' %$ $- | #$ 1 2" $ « ( $ -% "»: }20,18,16,14,12,10{=Γ }2010|{ ≤≤=Γ xxx µεςφυσικαρτιοςναιε
  2. 2. *. ,"& ! 2. & $ ! $% ! 3' " # / & !%"$ $ % $ - . "! $: % & . %$ - $ - : ( % ' $ % " $ 1 " $ $ 0) % " !& $ - : % & - $ - : % & / $ - $ - R " $ 1 "$ $ $ 4 $ 1(1 $ " !. , # # % #" " $ 1 "$ % $'"! : {}=∅ *. ,"& ! 2. & $ ! $% ! $% / " - ( % 1 $% /$ "$ !2 " $ ( % $'"! "$ #" "$ !% $' %" ( % : + . " $ #$ 1 2 " $ 5 "$ % % &N∈5+ . " $ #$ 1 2 " $ 5 "$ % % & . %$ - $ - - / . " $ #$ 1 2 " $ 4.4 #" "$ % % & . %$ - $ - % $ . ! /$ ( % "! $ $ % $'"! " $('"$ ( $ '$ %"$ " ! " $('"$) 3 %$ % $ N∈5 N∉4.4 }3,2,1{=A }2,1,3{=B$ ! $ !#$ ( . " $(' !#$ % $'"! ) - % "! $ % ( % % % , . " % $'"! ( "$ " $('" $ $1- ! . % % . " - + "! $ $1- !#$ % " % $ * 4 $ !2 " $ =*=+ ( % "! $ !% ) }2,1,3{=B }3,3,2,2,2,1,1{=Γ *. ,"& ! 2. & $ ! $% ! % $ "! $ % /( $ "$ ( & . % ! "$ $ " $ 1 $ "! " $ # " . # " " $ #"!/ :#"!/ : 3 % $ - : 3 % % 1 %"$ - : 3 % % %"& : ,...}13,11,7,5,3,2{=A },,,{ patjimjohntom=Β }1)(,1)(,)({ 4 32 ++=+===Γ xxxhxxgxxf 3 % % & ! 3 % " " % $'"! : }1)(,1)(,)({ ++=+===Γ xxxhxxgxxf },},5,2{},3,2,1{{ Ν∅=∆ }}5,1{,7,5,,2,14.3,{ tom∅=Ε *. ,"& ! 3. '(%"$ & 1. % !2 " $: ! !" # ! $BA ⊆ #"!/ : %' "$ $: %' "$ $: %' "$ $: " $%' "$ $: " $%' "$ $: !" # ! $ %& BA ⊆ },,,{},,{ dcbacba ⊆ N⊆}3,5,1{ }3,2,1{}3,2,1{ ⊆ }3,2,1{}4,3,2,1{ ⊆ }3,1{}2,1{ ⊆" $%' "$ $: $ $% %'(% % "! $ ") : & " ' % % !2 " $ $% % & : }3,1{}2,1{ ⊆ BxAxBA ∈∈∀⊆ καιειισχανν ABBABA ⊆⊆= καιανν
  3. 3. *. ,"& ! 3. '(%"$ & 2.+ %$ % !2 " $: " !" # ! $BA ⊂ #"!/ : %' "$ $: %' "$ $: %' "$ $: " $%' "$ $: " $%' "$ $: !" # ! $ ' % ( BA ⊂ },,,{},,{ dcbacba ⊂ N⊂}3,5,1{ }3,2,1{}3,2,1{ ⊂ }3,2,1{}4,3,2,1{ ⊂ }3,1{}2,1{ ⊂" $%' "$ $: $ $% %'(% % "! $ ") : }3,1{}2,1{ ⊂ Β∈Α∉∃⊆⊂ xxBABA καικαιανν *. ,"& ! 4. $ 5# (' " #"$ % " $(' " " % ( % $'"!& , & .'. =Β ! ('"$ 4 % $'"! . & % $'"!& % 2" $ $ $ $ % $ % 1 !2" $ " |*| ( % (% %" # " " / ( ). + . ": "$# & % ('"$ " " % ( % $'"!& , ( " $ "! $ ( " " % ( % . ! " " % " $(' "$ % $'"!& },,,{ patjimjohntom=Β 4|| =Β ! " " % " $(' "$ % $'"!& ! 2 $ "$ % "$ % , & .'. % $ - N,Z,Q,R " ' % - & % 1 $% - (' ": |||| BABA ≤⊆Α τετν |||| BABA <⊂Α τετν *. ,"& ! 5. )"$ & !2 " - )"$ " ! & % & . / " $ ( & %" # % $ *: ( &% #! "$ % " $ 1 "$ % $'"!BA∪( &% #! "$ % " $ 1 "$ % $'"! & $ * #! "$ % " $ 1 "$ $ % $'"! & $ * % & #! "$ % $'"! #" % #$ . #! "$ % $'"! % $ #" % * 4 $ $ " ! " )"$ #! ( % "! $ % $ BA∪ BA∩ BA − 4 $ $ " ! " )"$ #! ( % "! $ % $ % % ('"$ : "%$ /$ " # % & $ *: # % " % AxB )(AP *. ,"& ! 5. )"$ & 1.3 &% & !2 " $: ) * + , " #$ / Venn, & % "$ !2" $: ) * + , }|{ BxAxxBA ∈∈=∪ A B +$ #"$/ ={0,1,2,3} $ *={2,3,4,5} " }5,4,3,2,1,0{=∪ BA
  4. 4. !2 " $: *. ,"& ! 5. )"$ & 2. & ) ! , " #$ / Venn, & % "$ !2" $: ) ! , }|{ BxAxxBA ∈∈=∩ και A B +$ #"$/ ={0,1,2,3} $ *={2,3,4,5} " }3,2{=∩ BA *. ,"& ! 5. )"$ & 2. & % " $ # % : ! " " $(' $ % $'"! , #" "! $ "! " " $(' $ % $'"! , #" "! $ " % . # . $%' "$: $ ! " #" " $(' $ % $'"! , "! $ " ∅≠∩ BA 0|| ≠∩ BA A B ! " #" " $(' $ % $'"! , "! $ " % # . $%' "$: $∅=∩ BA 0|| =∩ BA A B " ! &% (/ $: «)( " ) % » *. ,"& ! 5. )"$ & 2. & ( ' "/ "$% – "$% ) ' "/ "$% – "$% "! $ ( $ %'" !2"$ $ ( &% $ # % & .%'" !2"$ $ ( &% $ # % & . / " $ ( $%' "$: ! /" $ " " $ $ /$ ! % & ") : |||||||| BABABA ∩−+=∪ |||||||| CBACBA ++=∪∪ || |||||| |||||||| CBA CBCABA CBACBA ∩∩+ ∩−∩−∩− ++=∪∪ *. ,"& ! 5. )"$ & 3. & !2 " $: ! - ! . , " #$ / Venn, & % "$ !2" $: ! - ! . , }|{ AxxA ∉= A U +$ #"$/ ={0,1,2,3} $ U={0,1,2,3,4,5,6} " }6,5,4{=A
  5. 5. !2 " $: *. ,"& ! 5. )"$ & 4. $ . & ) / -% , " #$ / Venn, & % "$ !2" $: ) / -% , }|{ BxAxxBABA ∉∈==− και A B +$ #"$/ ={0,1,2,3} $ *={2,3,4,5} " }1,0{=− BA {4,5}B A− = *. ,"& ! 5. )"$ & 5. 4 "%$ +$ " !2 " $: ! - ! , % ' $ " $ " (%"$ % % % (x,y) " " #$ " / ( 2" / " - % $'"! x $ #" " % $'"! y, %"$ "$ $% & " - 2" / ('"$ % %! (%" ! "% " % 1 $% " / $% "$ !2"$ $ %"$ "$ $% & " - #" ('"$ % %! ) - ! , }|),{( ByAxyxAxB ∈∈= και " - #" ('"$ % %! ) +$ #"$/ ={1,2} $ *={a,b,c} ": $#$ /$ $ "%$ /$ ( $%' "$: )}2,(),1,(),2,(),1,(),2,(),1,{( )},2(),,2(),,2(),,1(),,1(),,1{( ccbbaaBxA cbacbaAxB = = ||*|||||| BABxAAxB == *. ,"& ! 5. )"$ & 6. % !2 " $: " ! . !" # ! 0 1 2$ $ , ={1,2} " # % "! $ % : - ={1,2,3} " # % "! $ % : ! . !" # ! 0 1 2$ $ , + - * }|{)( Α= τουνολουποσναιεxxAP }}2,1{},2{},1{,{)( ∅=AP - ={1,2,3} " # % "! $ % : , #"!) " %" " " /$ $ $ " % : }}3,2,1{},3,2{},3,1{},2,1{},3{},2{},1{,{)( ∅=AP || 2|)(| Α =AP +. % %"$ 6% % 4 % 1 )" % " $ & %"$ . % $%' #" $%' : ∅∉∅ ⊂∅ ⊂ ⊆ ∈ Ν∈ .6 }2,1{.5 }2,1{}2,1{.4 }2,1{}2,1{.3 }2,1{}2,1{.2 5.1 RQZN RQZN ⊆⊆⊆ ⊂⊂⊂ ∅∉∅ ∅∉∅ .9 .8 }{.7 .6
  6. 6. +. % %"$ 6% % 4 % 2 4 % " % " #$ / Venn "$ !2 % : .1 BA∩ )()(.4 )()(.3 )(.2 .1 BABA ABBA CBA BA ∩∪ −∪− ∪∩ ∩ +. % %"$ 6% % 4 % 3 4 % " % " # % & % & : }4,3,2,1{.1 =A }}2,1{,1,{.2 }4,3,2,1{.1 ∅= = B A +. % %"$ 6% % 4 % 4 ( #"!/ 150 . $ - " " $% $ $ 50 (' # -%"$ & #$ $ - $ - , $ 80 (' # -%"$ "$% /&/ % . $ , " - 40 #" "' # -%"$ ("$% /&/ % . $ , " - 40 #" "' # -%"$ ( # . 1 " "! % $ . $ ( (' # -%"$ $ & #$ $ - $ - $ "$% /&/ % . $ .

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