Παρασκευή 1 Νοεμβρίου 2013 Ασκηση 1. Λύση. Παρατήρηση. Ασκηση 2. Λύση.

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< N = = = = =. = = () () = = {,,..., }, N. ( ) ( ) =, = = ( ) = = = () = ( ) = =, ( ) = ( ( ) ) =, ( ) = =, () =. (, ), ( ) = = ( ) =, ( ) = ( ( ) ) = ( ) = ( ) ( ) =., =, ( ) =, ( ) =, = (, ) (U(), ) U() = { : = = } (, ) (U(N), ) (U(Z), ) (U(Z, ) (N, ) (Z, ) (Z, ) (Z Z, ) (, ) (, ) = (, ) (U(Z Z), ) U(), U() & : = = & = = ( ) ( ) = ( ) = = = ( ) ( ) = ( ) = = = U() U() U() U() (U(), ) U() U() = = (U(), )

U(N) = {} N N = = = U(Z) = {, } Z Z = = = = U(Z ) = {[] Z & (, ) = } Z = {[], [],, [ ] } [] U(Z ) [ ] Z [] [ ] = [] = [ ] = [] = = = + Z (, ) = (, ) =, Z = + = [] = [ ] + [ ] = [] = [] [ ] + [] [ ] = [] = [] [ ] Z [] U(Z ) U(Z ) = φ() φ Z Z (, ) Z Z : (, ) (, ) = (, ) = (, ) (, ) (, ) Z Z (U(Z Z), ) (, ) U(Z Z) (, ) Z Z (, ) (, ) = (, ),,, (, ) = (, ) = = & = = = ± & = ± U(Z Z) = { (, ), (, ), (, ), (, ) } (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) = {,,, } :

= = V V (Z, +) Z Z Z Z = { ([], []), ([], []), ([], []), ([], []) } : + ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) ([], []) Z Z Z Z = =

(Z, +) (Z, +) + [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] Z Z = (, ) = {,,, } =, = = Z Z : = ([], []) = ([], []) + ([], []) = ([], []) = ([], []) = ([], []) + ([], []) = ([], []) = ([], []) = ([], []) + ([], []) = ([], []) =, = GL (R) = { GL (R) Z } = { GL (R) Z } (, ) (, ) =, =, =, : = {, } = {,,, } = = = = + = + = = = = GL (R)

(Z, +) (Z, +) : + [] [] [] [] [] [] [] [] [] [] [] [] + [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] Z Z : [] = {[]} [] = {[], [], [], [], [], []} = [] [] = {[], [], []} = [] [] = {[], []} Z : Z [] = [] [] 2 2222222 [] Z = [] = [] [] [] Z = {,,,,, : Q \ {, } Q \ {, } } () =, () =, () = () =, () =, () = (, )

: = Id Q\{,},,,,, : =, =, =, =, = (, ),, : = Id Q\{,}, = Id Q\{,}, = Id Q\{,},, = Id Q\{,} (, ) : ( ) ( )() = ( ()) = = = = () ( ) ( )() = ( ()) = = = = = = = () = = (, ) = = = = ( ) = () () : ( ) = = = = ( ) = () () = ( ) [ ( ) ] = ( ) ( ) = [ ( ) ( ) ] = [ ( ) ] () = ( [( ) ] ) = () = ( ) = () = = = =, ()

() : = = ( ) = = = ( ) [ ( ) ] = ( ) () = [ ( ) ] ( ) = () = ( ) = () = =, : = = () () () (, ) (, ) = =,, (, ) = =, = = = = = : = ( ) = ( ) = = = =, () = =, = () () () (, ) = R R :, (, ) (, ) = (, + ) (, ) = { : R R () = +,, R, } (, ), (, ), (, ) = R R : : (, ) [ (, ) (, ) ] = (, ) (, + ) = (, + + ) [(, ) (, ) ] (, ) = (, + ) (, ) = (, + + )

: (, ) (, ) (, ) = (, ) = (, ) (, ) (, ) (, + ) = (, ) = = + = = (, ) = (, ) = R R = = = (, ) (, ) = (, + ) = (, ) = (, + ) = (, ) (, ) (, ) = (, ) : (, ) (, ) (, ) (, ) = (, ) (, + ) = (, ) = = + = = = = (, ) (, ) = (, + ) = (, ) = (, ) = (, ) (, ) (, ) (, ) = (, ) (, ) (, ) (, ) = (, ) (, ) = (, ) (, ) :=, : R R, () = + : : :, = Id R : R R Id R () = = + :,,,, =, R :, (, () ) = =, ( + ) = = ( + ) + = = + + = = + = = = = = (,, ()) =, ( ) = + = =,, =,,, =,, (, ) =, (, )

: (, ), () = + (, ) (, ) = (, + ),, =,+ (, ), = Id R (, ) = (, ) (,) =,