www.touscours.net, Groupes,Permutations,Anneaux,Arithmétique dans Z,Corps commutatif,Les polynômes formels à une indéterminée à coefficients dans un corps K,Fonctions polynomiales,racines,Espaces vectoriels,K-algèbres,Espaces vectoriels de type fini,Matrices,Déterminants,Fractions rationnelles,Produit scalaire sur un R-ev,Espace vectoriel euclidien,R-ev euclidien orienté de dimension 2,R-ev euclidien orienté de dimension 3,Espaces affines,Géométrie dans un espace affine euclidien

1. ! " # ε $
%
& '
# "(= ) %
*
• ="( " %
+#
"("(
→×εε
, $ +
, =⇔= -"(
, "("( =
, "("("( +≤
• . ε " ε "
{ }∈= ""("(
• . &" / ε "
{ }&&&& """("( ∈∈=
0 1 # " # "2
3 ,
%
4/
. &" / &" " #
5"( & # 5"( & %
' 3
6 , # ε∈ -≠ " #
{ }+
∈+ λλ "% %
+# / , #
%
touscours.net

2. &
7 8
. , ε %
. ε∈ %
+ 8 = # 8
9 # 3 3 ⊥ ( : ,$, 3
⊥
∈ ) %
;
+ 8 # 3 3
"("( = %
< 8 %
∈ "
&&&
+= %
≥ " = = %
4 ∈ " (
∈
= %
> + =
• +#= ε∈ $ { }-?∈ " #
{ }⊥∈= "ε
• @ ' 3 A
+ #
⋅
→ε " ) ε∈ { }-?∈
∈ " # ⋅ " #
{ }=⋅∈= "ε %
< "( 3 ∈ % 4 "
3 &
%
= % " ε∈
-=⋅⇔
⋅=⋅⇔
=⋅⇔∈
#= $ %
•
B= /
. ε∈" " %
. { } { }=∈==∈= ""("(" εε
. C"D %
&(((
&&
&&
⋅=+⋅−=−=−
-=⋅⇔=
touscours.net

3. *
{ }-" =⋅∈= ε " E #=
$ % < # #= %
• # $ = %
. = $ %
. ε∈ F ="( ) 8 %
%λ
+= %
&
%λ=⋅ %
⋅
==
&
%λ
G +
. εε → %
ε ⇔ " : ,$,
"(("(("" =∈∀ ε
+ ε / ε ε
$ ( ( : ,$,
• . (∈ϕ " "
ε∈" " #" #
"((:::":( ==== ϕ
• . 3 %
< 3 # % . +=ϕ %
H 3 ϕ ( : ,$, 3 (∈ϕ
. ∈ " ε∈ " += % =
===== ::::(( ϕϕ ( I
! ε = ε " : ,$,
$ ( %
! ε = ε " : ,$,
$ (?( %
(J ε " ε " ( ,
(ε " # (ε ε , %
4/
+ = (J ε %
touscours.net

4. 0
4 " = $ , "
" = $ ( )
% 4 " 3 ⊥
= " 3 =
$ F =
$ " 3 # (
. =( " =( " " = $ "
3 − " %
+ / ( = $ =
%
4 / ε " / / 3
" # / #= #= %
(!
+ &
+
4
. " +=ϕ ( " (ε∈
(∈ϕ % < 3 ϕ " # %
• ϕ # " # # %
" %
' 3 " (ε %
• / ϕ # θ ( π& %
' %
. ε∈ " %
. ε∈ %
(:(( ϕ+=+= %
:(J(
:(:(/
=−⇔
=−⇔=⇔
ϕ
ϕϕ
<
{ } { }-("JK ( ==∈=− ϕϕ
ϕ 8 " 8 (
/ :(J( −
−=⇔ ϕ %
< / Ω % ($ :(J( −
−+=Ω ϕ
H ((("
:
Ω+Ω=Ω+Ω=∈∀ ϕε
" (: Ω=Ω ϕ %
touscours.net

5. 7
< 3 Ω # θ % J "
" # 3 Ω Ω
(∈θρ %
' 3
3 (:= $
=ΩΩ
Ω=Ω
C&D:5"(
:
πθ
(ε
4 1
∅ ;
ε J
;
' { }Ω ' #
Ω
& /
. / " & / & %
. &= " εJ& = %
. &≠ &LL " =& " ) &&=
.
Ω
θ
& Ω # / #
θ& )θ # 5"( &
(J 8 #
J " L " /
L Ω
3 / / L %
" / / %
; / " M -" " &" *
/ %
4
. (J ε∈ %
. " 3 # - & / %
. / 3 3 " (ε∈ "
- & / " −
= *
/ %
touscours.net

6. >
* +
. (?(J εε∈ %
+=ϕ % (?(∈ϕ %
ϕ / " (N= %
/ %
" ε∈ " "
(((: ϕ+=+==
(: ϕ=
/ %
J " / (?(J εε
& # / %
. ε∈ " (:= %
< 3 %
" ϕϕ =J %
= /
:
= %
H " : #
⊥
∈∈
+=: % == :
%
(
(
< = % ϕϕ == J+ % H / (
" / &+= % /
LL: &+ (
" = " ) /
@ A $ / %
(:=
:
(
touscours.net

7. O
! /
% < = / %
4 / 1
∅ ;
ε εJ
{ }Ω ' # Ω
' / %
J
∅ ' / ( : ,$, )
/
{ }-?(∈
G = 3 &
. ""(= " %
! 3 =+ " ) -"-("( ≠ % !
$ " −
%
# $
.
-
-
- " =+ % <
. ∈ %
&&
--
&&
---
-
((
"(
+
−+
=
+
−+−
=
⋅
= %
&&&& =+
=+
θ
θ
+# θ " # # 5"( & %
&
&
&
&
&&
&&
&
&
++
+
=
⋅
=θ
43
. "
&
&
& %
+#3
-
&
((
&
((
--
&
&
&
& =
+
−−+
+
−− %
touscours.net

8. +
. εε → %
⇔ "(%("(("""P =∈∀∈∃ + ε %
"
. " "(%("(("""PQ =∈∀∈∃ + ε %
%
. εε → " -> %
< 3
⇔ #
# %
⇔ #
ϕ% ) (∈ϕ
⇔
%
"
. % . ψ %
(∈= ψϕ %
• . ϕ ( " 3 " 3
%
• " &
(
• . " " %
+# ε , "(( ε %
& 4 /
< )
""-( %
R 3
• ; ) ∈ %
!! +
• .= ( / $ #/
!!
touscours.net

9. • B ∈α
!! %α %
B -! ∈α
%( -- !!!! −+α ( ---- %( α+=+
• ' # ∈θ
!"! θ%
' -! # ∈θ
( -
%
- !!"!! −+ θ
+
+ / =
!! +% " ) ×∈ P"( %
• . ×∈ P"( %
# θ
α %
%" " ) P∈α ∈θ %
!! +%
!"! θ%
(
α+ (
4 %α (
# # # "
%
• J " %
4 # # ( : ,$,
# " =
!! +% 3 # =
!! +% %
4
. !! +% %
-! / !!! =−⇔=+⇔ --- (%
. = -≠ " #= / " !! +
%
. = -= " # %
. ≠ " / -! %
!! += %( " !! += -- % %
(( -- !!!! −=− %
# θ
α %
%" ) P∈α ∈θ " 3 "
" # -! α
-! # θ %
J 3 !! +% " )
×∈ P"( %
touscours.net

10. -
4 " # !! +% "
# !! +% "
%
J " " !! "
" " #
+% " = " !! +% %
/
4 1
∅ ;
J
{ }-! . $ " : ,$,
( #
-! # θ #
-! α ( % %
≠θ
α "
('
. C"D C:":D /
( / %
/ 3 C"D
C:":D (
" / " " " % . ∈βα" %
. βα +!! % %
=+
=+
:
:
%
%
!!
!!
βα
βα
" : ,$,
−=
−=−
!!
!!!!
%
%(
:
::
αβ
α
%
< 3 P∈α " ∈β %
=:: %
. " $
::
#
::"(
!!
!!
−
−
= ::
α "
!!
!! ::::
=
−
−
=α
C&D::"(
"(::"(
((( ::
π
α !!!!
=
−=
−−−=
touscours.net

11. . ε %
. ""(= ε %
. ε∈ &
"( ∈θρ %
< 3 "( θρ =
3 (% θρ= " ) (θ %% θθ + " : ,$,
3 C&D(5"( πθθ = %
J 3 # 8 =
"
• + = / "-( θρ = =
%
• ! ≠ / =
∈++−=∈+= "&"(""&"( ππαρπαρ " ) α
# "( %
43 " /
• + #3 *=ρ ( $ #
ε∈ 3 # = "( θρ
*=ρ % " # # ε∈ 3 # /
∈θ 3 (%* θ= " # = *%
• #3 θρ = " +
∈θ ( ∈θ
π&
π
&Lπ
&L*π
J " ε *" #
"""(= %
(+ *
+
4
. (ε∈ " ϕ %
touscours.net

12. &
(∈ϕ %
# " #/ "( ω # θ %
• . C&D- πθ = " J=ϕ " %
• . C&D- πθ ≠
, . # / # " 3 %
ε∈ " ( 3 (: ϕ= %
. " 8
%
∈ " ∈ " ϕ "
( =: %
(: ϕ= %
< 3 #/ "( ω # θ
ω
θ
# / %
J " = %
, . # /
. ε∈ " %
:
= % ϕϕ =J
% # #/ "( ω # θ )
" θ # ϕ "
:
=
ω
θ
H : :
⊥
∈∈
+=: %
4 = %
. $ %
( / ϕ ⊥
: ,$,
(
M " $
" $ 3 ϕ # # % <
/ L % / % #
#/ ":( ω # θ ) # %
touscours.net

13. *
= " ) #/ ":( ω # θ "
%
ω
θ
# M $ #( -≠ "
#= / %
< 3 ( #/ ":( ω " # θ %
(
4 1
∅ J ;
ε J εJ
θρ " -≠θ #/ "( ω
) (=
' #/ "( ω # θ
∅ θρ " -≠θ #/ "( ω N #/ "( ω
# θ " )
" ∈≠ -
3 " /
' E
:
:
:
:
!!
)
+=
−=
−=
&:
&:
>:
!!
( #
# # 3
&
&
>
:
# −
−
!!
: " : ,$, # ϕ
−
−
--
--
--
""( %
4 " 3
(:((" ϕε +=+=∈∀
. −
−
+=
!! --
--
--
&
&
>
:
:
:
touscours.net

14. 0
(& J
# # π ! ( =
$ ! " #/ !
# ( =ϕ (∈ϕ
% J≠ϕ " "
#/ %
(* '
!
=
=
=
⇔
+=
−=
−=
⇔
-&
*
&
&
>
!!
< # " # %
(0 #/ " #
. #/ "( ω # -≠θ -≠ %
{ }$:"ε∈= " # 3 : 3
%
$" −
−
&
&&
&>
: %
=
=
⇔∈
*
(N:
#/ !
-
*
%
/ /
, . &LL " =& " ) &&= ( #
&=
&
&
, .
α
=∩ & " & # α& %
' 3
! 3 / " /
& ( # " "
" %
touscours.net

15. 7
G = 3 *
43
#! =++ " ) -"-"-(""( ≠# %
! $
#
< 3 " ε∈
-=⋅⇔⋅=⋅⇔=⋅⇔∈
<) 3 %λ= &
=λ %
J /
&&&&& !#
!#
=++
=++
<
#
"
&
&
&
&
#
%
. -& =∧ " &LL %
. -& ≠=∧ " &∩ " %
4
⊥ " (∈ " &⊥ ( &∈ %
(( &∩=∈ ( -≠ %
# $
. $ %
%
="( " %λ= %
+= %
&&
%-% λλ =+=⋅
" &
⋅
= "
&&&
#
#!
++
−++
=
⋅
= %
# $
="( % < ∧=∧+∧=∧ %
=∧ ( ⊥ %
∧
=="( %
touscours.net

16. >
= 3 3
= 3
. ε∈ " ( ""( ! %
< = = 3
""( !$ θ *
""(""( !$$! θθ= %
" " 3 ∈θ " %%( θθθ += " "
3 ε∈ " % 8 "
3
""( !$ θ = = 3 $
⇔+=⇔ !$ %(% θ "( θ$ = %
""( ! S %
4 ε =
= 3
• . !∈ " = ""-( !θ " ∈θ 3 3 %
• . !∉ " # # /
$ = " 5"( %=θ ( "( " ! S %
4 "&"( !$ πθ + "&"( !$ ππθ ++− ∈
3 3 %
' 3
/ $ θ ( !∉ "
&&
$ += " $=θ " $=θ %
θ
& 3
. ε∈ " ( ""( ! %
< = 3
""( ϕθ$ *
ϕθ$= " ϕθ$= " ϕ$! = %
4
. ε∈ " ""( ! % < 8
% 8 %
3 ""( ϕθ$ = 3 $
3
( "( ϕθ$ = % $
""( θ$! = %
touscours.net

17. O
, . = " ( 3 -== θθ $$ " 3 -=$ %
' 3 " ""( ϕθ$ 3 -=$ =
3 %
, . ≠ " !∈ " ( 3 θ$! = -=θ %
' 3 " ""( ϕθ$ " : ,$, =
"&"( ϕπ! "&"( ϕππ! +− ( ∈ϕ 3 3
= 3 %
, 4 " !∉ "( αρ =
% ""( " ( 3
+=−==
===
C&D
C&D
ππαϕρθθ
παϕρθθ
$!$
$!$
' 3 " ""( ϕθ$ =
3 %
J 3 ε =
3 %
' 3
. ""( ϕθ$ = 3 " 8
&
&
$ =
4 &&&&&&&
&
( $$$ =++= θϕϕθ
' # = 3
!∉ %
. ε∈ " 3 !∉ 8 % 8
" ""( ! %
$ = %
. [ ]πθ "-∈ # "(
θ$! =⋅= %
" &
&&
!% += " θθ &&&&
&
( $$% =−=
" -≥$ -≥θ ( [ ]πθ "-∈ " θ$% =
. ϕ # "( "( %
%%($% ϕϕθ +=
4 ""( ϕθ$ = 3 %
ϕ
θ
touscours.net

18. ' 3
3 !∉ " ->$ ] [πθ "-∈ %
ϕθ""$
&&&
!$ ++= " =
$
!
θ "
θ
ϕ
$
= "
θ
ϕ
$
= %
" = 3 3
] [ [ ]ππϕπθ """-"- −∈∈>$
# # " = 3
:&"&"(
:&"&"(
:&"&"(
:&"&"(
ππϕππθ
πϕππθ
ππϕπθ
πϕπθ
$
$
$
$
++++−−
+++−
+++−
++
4 3 3 ε " 3 # #/ ! "
= 3 ""( ϕθ$ -≥$ " [ ]πθ "-∈ C"D ππϕ −∈ %
H $ = "
3 3
( / %
touscours.net