1. !!" #= $ !
→ % !
& !!!# % %
=
=∈∀ ""
"& % $
' %
( ∈ ∈ !!!# !
[ ]
)"
""))
=−⇔
=∈∀⇔
<−<−∈∀>∃>∀⇔= εααε
' % * ∞+ ∞− +
, !
' %
+ ] [+∞∩ ] [∞−∩ !
& $ + %
→ ∈λ
→ϕ % !
+ λ ϕ ⋅ +
+ λ !ϕ ⋅ !
- . / ∧ +
∧
& 0
( →ϕ " ⊂ → " ⊂"ϕ ∈α
ϕ ∈ α
ϕ α + !
' %
( ∈ !
⇔ " * "
[ ]∈∀⇔ !
"' % * 1
touscours.net

2. #
' %
∈∀⇔ !
[ ]∈∀⇔ !
& , %
+ % 2
& % 0
( → $
" 3 + * 4 ∈ + ≤∈∀ "
& %% *+
$ [ ]∈∀⇔ $ !
5 % $ =
" %
' %
( ∈ !
$ ⇔ * { }
−
−
→
""
6 !
[ ]∈∀⇔ $ !
"3 "
=
= "3"3 !
' % * $
$ !
"7
( $ )3 =⇔=
8 % 4 %
% 2
9 & % $
( → $ ∈λ →ϕ $
+ λ ϕ ⋅ $
"3"""3"3"
"3!""!"3"3!"
"3!"3!"
"3"3"3"
⋅+⋅=⋅
+=
=
+=+
ϕϕϕ
λλ
- . / ∧ $
"3"""3"3" ∧+∧=∧ !
touscours.net

3. /
7 +
& $ % : $ ; ""
"!" ⋅ϕ
"
" ∧ + ϕ !
< 0
( →ϕ → ⊂"ϕ ∈α ϕ $ α
$ "αϕ ϕ $ α ""3"3"3" αϕαϕαϕ =
( → $ )" ≠ $
"
"3"
"3"
⋅
=
- %% ⋅= + 0
% ⋅
( )"" ≠⋅ " 3 )" ≠ ⋅ $
"
"3"
""
"3"#
#
⋅
=
⋅
⋅
!
( $ = 3⊥
" 3 "3" ⊥∈∀
- %%
( )== * + )== *. !
(
"
"3"
"3")
⋅
==∈∀
-4
* ! " # $
4 $ "
& , % * "
∈ % !
$ !
!
: + * = % !
"! #
π+
= $ !
%
( →=> !
=
"
4 $ !!!" #= ! % " !
: % * 4% 42
touscours.net

4. ?
2
< 0 "
( →=> " 4 ≤
≤ ""
< 0
( → % "
* * + !
5 + →
""" −= !
+ 0 * $ 2
7 + : % % " = 4 !
!
5 %
( => $ >= 4 +
∈ +
≤∈∀ "3>= −≤− "" !
:* % % " =
5 < @: * −
( " ≥ => ""
=>∈
= "+ 4 * 0
A
"
A"
"
!!!"3"""" " −
≤
−
−
++−+− −
−
:* < @: % "
< " * −
→=> " ≥
−
−
+
−
−
++−+=
−
−
−
"
A"
"
A"
"
!!!"3""" ""
" % * $ B
* < @: B 0 ,
< @C " *
( → ) 4 →ε
+ )"
)
=ε
")"!!!!)"33!)"3!)"" "
A#
#
ε+++++=∈∀
touscours.net

5. ' %
( % ∈) = { })6 → !
& + ': * ) + * 4 % →ε
!!!) +
"""!!!"""
)"
))#
#
)))
)
ε
ε
−+−++−+−+=∈∀•
=•
D * ': * ) !
-4 * ': * ) 4 ': ) ≤ !
-4 * ': * ) ) ⇔ 4 * )
- + ∈) " 3 + =
-4 ': * ) ⇔ $ )
"E * 4
) ': * )
"' % < @C
& ':
( !
+ + !
touscours.net