exercice_logique_1sc

					‫ز‬   ‫ا‬   ‫را‬   ‫ا‬                         ‫ا‬              :‫ذ‬       ‫زا‬           ‫إ‬              ‫م‬                 ‫ىا و‬                           ‫ا‬   ‫دئ‬       :   ‫ر‬
                              Chorfi_mouhsine@yahoo.fr
                                                                                                                                                 :1 ‫ر‬
                                               : ‫رة‬                     ‫آ‬                             ‫د‬      ‫و‬        ‫ت ا رات ا‬                 ‫لا‬            ‫أآ‬
                                ( P ) : n = 2m :                       m                                     ‫د‬           ، n                       ‫د‬         **
                                               (Q ) : x ≤ M :                                 x                  ‫د‬                   M           ‫د‬           **
                                            ( R ) : x 2 − mx + 1 = 0                                        x             ‫د‬             m       **   ‫د‬
                                                                                                                                            : ‫ا اب‬
                                                                                                          ( P ) : ( ∀n ∈ ℕ )( ∃m ∈ ℕ ) : n = 2m **
                                                                                                   ( Q ) : ( ∃M ∈ ℝ )( ∀x ∈ ℝ ) : x ≤ M                      **
                                                                                        ( R ) : ( ∀m ∈ ℝ )( ∃x ∈ ℝ ) : x 2 − mx + 1 = 0                      **
                                                                                                                                               :2 ‫ر‬
                                                                                          :                 ‫آ‬                 ‫د‬     ‫و‬      ‫ا رات ا‬  ‫د‬
                                                                                                                                  ( P ) : 13 ≥ 5 + 8 (1
                                                                                                                    ( Q )( ∃x ∈ ℝ ) : x 2 − x + 2 = 0 (2
                                                                                                                 ( R )( ∀x ∈ ℝ )( ∃y ∈ ℝ ) : x < y (3
                                                                                                          ( S ) : ( ∀x ∈ ℝ )        1 + x 2 − x ≥ 0 (4
                                                                                                                            (T )( ∀x ∈ ℝ ) : x 2 > 0 (5
                              Chorfi_mouhsine@yahoo.fr
                                                                                                                                                : ‫ا اب‬
                                                                                                                              ( 7 P ) : 13 < 5 + 8 (1
                                                (              )                                             ( 13 )
                                                                   2                                                  2
                     5 + 8 > 13             ‫و‬        5+ 8              = 13 + 2 40 ‫و‬                                      = 13 : ‫ت‬                   ‫رن ا د‬
                                                                                                  .                ( P) ‫و‬              ( 7P ) ‫إذن ا رة‬
                                                                                                                 ( 7Q )( ∀x ∈ ℝ ) : x 2 − x + 2 ≠ 0 (2
                                                                                    ‫د‬      ‫ا د : 0 < 7− = 8 − 1 = ∆ إذن ا‬
                                                                                    .          (Q ) ‫و‬            ( 7Q ) ‫ا رة‬                                 ‫و‬
                                                                                                   ( 7 R )( ∃x ∈ ℝ )( ∀y ∈ ℝ ) : x ≥ y                           (3
                         .( x < y :                 y = x +1                    ‫د‬             x           ‫د‬       ‫) ن‬           ( R ) ‫رة‬                             ‫ا‬
                                                                                             ( 7 S ) : ( ∃x ∈ ℝ )      1 + x2 − x < 0                            (4
                                (          1 + x2 ≥ x2 = x :                               x         ‫د‬        ) :‫ن‬              ( S ) ‫رة‬                             ‫ا‬
                                                                                                              ( 7T )( ∃x ∈ ℝ ) : x 2 ≤ 0                         (5
                 .       ‫رة‬     (T )   ‫رة‬       ‫ا‬      ‫: 0≤0 و‬                            x=0          ‫د‬                     ( 7T ) ‫رة‬                           ‫ا‬
                                                                                                 :3 ‫ر‬
                                                                   .                             ‫لا‬
                                                                                              ‫رة ا‬           ‫د أن ا‬           ‫لا‬        ‫ل‬
                                                       ( P ) ( ∀n ∈ ℕ* )( ∀m ∈ ℕ* ) : +
                                                                                     1      1           1
                                                                                               + .... + ∈ ℕ
                                                                                     n n +1             m
                                                                                                       : ‫ا اب‬
                 ( 7 P ) ( ∃n ∈ ℕ* )( ∃m ∈ ℕ* ) : +
                                                 1   1             1
                                                          + .... + ∉ ℕ :              ( P ) ‫** د ا رة‬
                                                 n n +1            m
                                 1 1 3
                                . + = ∉ ℕ : ‫7 ( ن‬P ) ‫ن ا رة‬                ‫ . أن ا د‬m = 2 ‫ و‬n = 1 :
                                 1 2 2
                                                                 .     ‫ ( رة‬P ) ‫و‬             ‫7 ( رة‬P ) ‫إذن‬
                                  Riyadiyate.site.voila.fr
                                                                                                            :4 ‫ر‬
                                                                                      : ‫أن‬          ‫د‬    ‫ام ا‬‫ل‬   ‫نا‬
                                                                                                         x− y
                                                                      . ( ∀x ∈ ℝ )( ∀y ∈ ℝ ) : x ≠ 0 ⇒         ≠ −1 **
                                                                                                         x+ y
                                                                                              x             y     
                                            . ( ∀x ∈ ℝ )( ∀y ∈ ℝ ) :  xy ≠ 0 ‫و‬x ≠y ⇒ 2               ≠ 2           **
                                                                                          x + x +1 y + y +1 
                                                                                                                : ‫ا اب‬
                                                                                           x− y
                                                          ( ∀x ∈ ℝ )( ∀y ∈ ℝ ) : x ≠ 0 ⇒         ≠ −1 : ‫أن‬          ***
                                                                                           x+ y
                                                                                      x− y
                                                             . ( ∀x ∈ ℝ )( ∀y ∈ ℝ ) :       = −1 ⇒ x = 0 : ‫أن‬
                                                                                      x+ y
                                                                       x− y
             x=0 :         ‫2 و‬x = 0       ‫ و‬x − y = − x − y ‫إذن‬              = −1 :                ‫ د‬y ‫ و‬x
                                                                       x+ y
                                              x− y                                      x− y
              ( ∀x ∈ ℝ )( ∀y ∈ ℝ ) : x ≠ 0 ⇒        ≠ −1 ‫∀ ( و‬x ∈ ℝ )( ∀y ∈ ℝ ) :             = −1 ⇒ x = 0 : ‫إذن‬
                                              x+ y                                      x+ y
                                                                                x             y     
                                ( ∀x ∈ ℝ )( ∀y ∈ ℝ ) :  xy ≠ 0 ‫و‬x ≠y ⇒ 2              ≠ 2            : ‫أن‬         ***
                                                                            x + x +1 y + y +1 
                                                       x             y                           
                             ( ∀x ∈ ℝ )( ∀y ∈ ℝ ) :         = 2             ⇒ xy = 0‫ أو‬x = y  : ‫أن‬        ‫أن‬
                                                    x + x +1 y + y +1
                                                           2
                                                                                                  
                                                              x               y
              :       ‫ و‬xy 2 + xy + x = yx 2 + xy + y ‫2 و‬             = 2           :                ‫ د‬y ‫ و‬x
                                                           x + x +1 y + y +1
xy = 1‫ أو‬y − x = 0       ‫ ( و‬xy − 1)( y − x ) = 0 ‫ و‬xy ( y − x ) + ( x − y ) = 0 ‫ إذن‬xy 2 − yx 2 + xy − xy + x − y = 0
                                                                                                . xy = 1‫ أو‬x = y ‫ا أن‬        ‫أ‬
                                      Chorfi_mouhsine@yahoo.fr
                                                                    x        y                                              
                                         ( ∀x ∈ ℝ )( ∀y ∈ ℝ ) :  2       = 2       ⇒ xy = 0‫ أو‬x =                         y  : ‫إذن‬
                                                                 x + x +1 y + y +1                   
                                                                                        x       y     
                                             ( ∀x ∈ ℝ )( ∀y ∈ ℝ ) :  xy ≠ 0 ‫و‬x ≠y ⇒ 2       ≠ 2                                     ‫و‬
                                                                                    x + x +1 y + y +1 
                                                                                                 :5 ‫ر‬
                                                                                       : ‫ت‬   ‫ا‬   ‫ل ن‬                               ‫ا‬
                                                                   . 2 x −1 = x :   ‫ ا د ا‬ℝ        ‫ا‬                             (1
                                                                   . n∈ℕ              n ( n + 1)( n + 2 ) ‫د‬    3 ‫أن ا د‬  (2
                                                                                                                     : ‫ا اب‬
                                     . ‫رات‬        ‫ول‬           x ‫ و‬x − 1 ‫د إ رة‬          ℝ      2 x −1 = x     ‫ا د ا‬     (1
                                                                                                                   : ‫أن‬
               ]−∞, 0] ‫ل‬     ‫ا‬                                      [ 0,1] ‫ل‬      ‫ا‬                        [1, +∞[ ‫ا ل‬
      x = −x ‫ و‬x −1 = − x + 1                             x = x ‫ و‬x −1 = − x + 1                            x = x ‫ و‬x −1 = x −1
 :    ‫ا‬            ‫نا د‬         ‫و‬             :         ‫ا‬              ‫نا د‬          ‫و‬          :       ‫ا‬            ‫نا د‬         ‫و‬
               2 ( − x + 1) = − x                                    2 ( − x + 1) = x                                2 ( x − 1) = x
                 −2 x + 2 = − x                                        −2 x + 2 = x                                    2x − 2 = x
                        − x = −2                                          −3 x = −2                                           x=2
                            x=2
                                                                               x=
                                                                                    2                     2 ∈ [1, +∞[ ‫و‬
      2 ∉ ]−∞, 0] ‫و‬                                                                 3
                                                       2
                                                          ∈ [ 0,1] ‫و‬
                                                       3
                                                 2 
                                             S =  , 2  ‫إذن‬
                                                 3 
                                         Riyadiyate.site.voila.fr
                                                                                           . n∈ℕ           n ( n + 1)( n + 2 ) ‫د‬                      3 ‫أن ا د‬            (2
                                                                                                                                         .          ‫ د‬n
                           . k ∈ℕ                    n = 3k + 2 ‫ أو‬n = 3k + 1 ‫ أو‬n = 3k :                       ‫ا‬                n              ‫د‬     ‫أن آ‬
                          :     ‫ا‬     ‫ا‬                                           :    ‫ا‬      ‫ا‬                                                : ‫ا و‬       ‫ا‬
    . k ∈ℕ         n = 3k + 2 : ‫إذا آ ن‬                   . k ∈ℕ           n = 3k + 1 : ‫إذا آ ن‬                             . k ∈ℕ          n = 3k : ‫إذا آ ن‬
          n ( n + 1)( n + 2 )     :‫ن‬                           n ( n + 1)( n + 2 )       :‫ن‬                                     n ( n + 1)( n + 2 )     :‫ن‬
       = ( 3k + 2 )( 3k + 3)( 3k + 4 )                       = ( 3k + 1)( 3k + 2 )( 3k + 3)                                     = 3k ( 3k + 1)( 3k + 2 )
       = ( 3k + 2 ) 3 ( k + 1)( 3k + 4 )                     = ( 3k + 1)( 3k + 2 ) 3 ( k + 1)                                   = 3  k ( 3k + 1)( 3k + 2 ) 
                                                                                                                                                           
      = 3 ( 3k + 2 )( k + 1)( 3k + 4 ) 
                                                           = 3 ( 3k + 1)( 3k + 2 )( k + 1) 
                                                                                                              .3‫د‬                          n ( n + 1)( n + 2 )          ‫و‬
.3‫د‬               n ( n + 1)( n + 2 )           ‫و‬    .3‫د‬                  n ( n + 1)( n + 2 )              ‫و‬

                                                                         . n∈ℕ                3‫د‬                    n ( n + 1)( n + 2 ) ‫أن ا د‬               ‫ا‬            ‫وأ‬
                                                                                                                                                         :6 ‫ر‬
                                                                                                                                                       : ‫أن‬
                                                                                                                n ( n + 1)( 2n + 1)
                                                                                  12 + 22 + ...... + n 2 =                                             ∀n ∈ ℕ*            (1
                                                                                                                            6
                                                                                                           . ∀n ∈ ℕ*                 9            ‫ا‬   10n − 1 (2
                                                                                                                                             ∀n ≥ 4: 2n ≥ n 2 (3
                                                         Chorfi_mouhsine@yahoo.fr
                                                                                                                                                               : ‫اب‬           ‫ا‬
                     n ( n + 1)( 2n + 1)            1(1 + 1)( 2 ×1 + 1)       6
                                                =                         =     = 1 ‫ + ...... + 22 + 21 و‬n 2 = 1 : ‫ ن‬n = 1 ‫: إذا آ ن‬                                 ** (1
                              6                              6                6
                                                                                              . n =1            ‫ول‬                 ‫رة‬                            ‫ا‬     ‫و‬
                                                     n ( n + 1)( 2n + 1)
                    . 12 + 22 + ...... + n 2 =                     : ‫أن‬       n ‫ــ‬                 ‫ض ا رة‬                                       . n ∈ ℕ*             **
                                                       6
               12 + 22 + ...... + n 2 + ( n + 1) =
                                                2  ( n + 1)( n + 2 )( 2n + 3) : ‫أن‬      n + 1 ‫ــ‬                                                 ‫رة‬     ‫أن ا‬         **
                                                                 6
                                                        (      ‫ا‬      ‫ا‬      n + 1 ‫ا ول ــ‬       ‫ا‬    n ‫ض‬                                      )
                              1 + 2 + ...... + n + ( n + 1) = (1 + 2 + ...... + n ) + ( n + 1)
                                                                     2                                                  2
                                2       2                2                        2       2            2
                                                                                                                                                                 :
                                        n ( n + 1)( 2n + 1)                               n ( n + 1)( 2n + 1)       6 ( n + 1)
                                                                                                                                 2

                                    =                            + ( n + 1)           =                        +
                                                                              2

                                                     6                                             6                   6
                                        n ( n + 1)( 2n + 1) + 6 ( n + 1)
                                                                                  2
                                                                                          ( n + 1)  n ( 2n + 1) + 6 ( n + 1)
                                                                                                                            
                                    =                                                 =
                                                            6                                             6
                                                ( n + 1) 2n 2 + n + 6n + 6
                                                                                         ( n + 1) ( 2n + 7n + 6 )
                                                                                                       2

                                            =                               =
                                                           6                                6
                                         :      ‫2 و‬n + 7 n + 6 ‫ا ود‬
                                                       2
                                                                                                 2n 2 + 7 n + 6 = 0 ‫ا د‬       ‫ل‬
                                           ( n + 1) × 2 ×  n +  ( n + 2 ) ( n + 1) × ( 2n + 3)( n + 2 )
                                                                 3
                                                                  
                                       =                        2
                                                                               =
                                                             6                                 6
                                                  12 + 22 + ...... + n 2 + ( n + 1) =
                                                                                    2   ( n + 1)( n + 2 )( 2n + 3)              :                                      ‫و‬
                                                                                                     6
                                                                   n ( n + 1)( 2n + 1)
                                        12 + 22 + ...... + n 2 =                                 ∀n ∈ ℕ*                      :                                               ‫و‬
                                                                             6
                                                                                    . ∀n ∈ ℕ*          9           ‫ا‬ 10n − 1 ‫أن‬                                        (2
(       ‫ول‬          ‫رة‬    ‫ا‬         ). 9                ‫ا‬      101 − 1 ‫ 01 و‬n − 1 = 101 − 1 = 10 − 1 = 9 : ‫ ن‬n = 1 ‫: إذا آ ن‬                                           **
                                       9              ‫ا‬      10n − 1 . : ‫أن‬           n ‫ــ‬                  ‫ . ض ا رة‬n ∈ ℕ*                                            **
                                                     . 9             ‫ا‬     10 n +1 − 1 : ‫أن‬        n + 1 ‫ــ‬            ‫أن ا رة‬                                         **
                                                         Riyadiyate.site.voila.fr
                                                                                10 n +1 − 1 = 10 × 10 n − 1 :
                           10n − 1 = 9k       k                 ‫د‬            9            ‫ا‬     10n − 1 ‫أن‬                        ‫و‬
                                                                                           . 10n = 9k + 1 :                       ‫و‬
               10n +1 − 1 = 10 ×10n − 1 = 10 × ( 9k + 1) − 1 = 90k + 10 − 1 = 90k + 9 = 9 (10k + 1) :                             ‫و‬
                                                                      .9          ‫ا‬    10 n+1 − 1                             : ‫إذن‬
                                          . ∀n ∈ ℕ*      9       ‫ا‬      10n − 1 ‫ن‬      ‫أا‬                                        ‫و‬
                                                                        . ∀n ≥ 4: 2 ≥ n 2 ‫أن‬
                                                                                    n
                                                                                                                                (3
(       ‫ول‬        ‫رة‬    ‫ا‬     ) . 2n ≥ n 2 ‫ و‬n 2 = 42 = 16 ‫2 و‬n = 24 = 16 ‫ ن‬n = 4 ‫: إذا آ ن‬                                     **
                                   . 2n ≥ n 2 : ‫أن‬      n ‫ــ‬             ‫ . ض ا رة‬n ≥ 4                                         **
                                        . 2 ≥ ( n + 1) : ‫أن‬
                                            n +1
                                                               n + 1 ‫ــ‬
                                                      2
                                                                                     ‫أن ا رة‬                                    **
                                  2 × 2 n ≥ 2n 2 ‫ن‬           ‫اض ا‬     ‫إ‬            2n ≥ n 2 : ‫أن‬    ‫ 2 و‬n +1 = 2 × 2n :
                                                                                                    2n +1 ≥ 2n 2 : ‫ن‬              ‫و‬
                                                                                   . 2n 2 ≥ ( n + 1) : ‫أن‬
                                                                                                    2
                                                                                                                ‫ا ن أن‬
                                        2n 2 ≥ ( n + 1) ⇔ 2n 2 ≥ n 2 + 2n + 1 ⇔ n 2 − 2n − 1 ≥ 0
                                                       2
                                                                                                                              :
    . 1− 2 ‫2 +1 و‬                        n 2 − 2n − 1 = 0 ‫ : 8 = ∆ إذن ا د‬n 2 − 2n − 1 ‫ا ود‬       ‫د‬
                                                         n − 2n − 1 ≥ 0 ‫ن‬
                                                          2
                                                                              ‫ و‬n ≥ 1 + 2 ‫ ن‬n ≥ 4 ‫أن‬
             . n +1 ‫ـ‬                        ) 2n +1 ≥ ( n + 1) ‫أن‬            2n +1 ≥ 2n 2 ‫2 إذن‬n 2 ≥ ( n + 1) :
                                                               2                                                   2
                             ‫رة‬    ‫ا‬                                                                                              ‫و‬
                                                                                                  . ∀n ≥ 4: 2n ≥ n 2 : ‫إذن‬
                                                                                            (         ‫ل‬     ‫ر 7 )ا‬
                                                                                        .          ‫ أ اد‬z ‫ و‬y ‫ و‬x      (1
                                                                                                2 x − 3 y > 3
                                                                                                
                                                                              .                 3 y − 2 x ≥ 3         ‫أن ا‬
                                                                                                 y−z≤2
                                                                                                
                                                           . P ( x ) = x 4 + 12 x − 1 :           ‫ود‬    ‫را‬         0 ‫أن‬  (2
                                                                                                                     : ‫ا اب‬
                                                                                  2 x − 3 y > 3
                                                                                  
                                                       ( x, y , z )               3 y − 2 x ≥ 3        ‫ا‬        ‫ض أن ا‬           (1
                                                                                   y−z≤2
                                                                                  
. 0<6          ‫: 6>0 وه ا‬              ‫ 2 ( و‬x − 3 y ) + ( 3 y − 2 x ) > 3 + 3 ‫ 3 إذن‬y − 2 x ≥ 3 ‫ 2 و‬x − 3 y > 3 :
                                                                                                2 x − 3 y > 3
                                                                                                
                                                                              .                 3 y − 2 x ≥ 3         ‫نا‬         ‫و‬
                                                                                                 y−z≤2
                                                                                                
                                              . P ( 0 ) = 0 ‫ إذا‬P ( x ) = x 4 + 12 x − 1 ‫ود‬              ‫ر‬    0 ‫ض أن‬              (2
                                                                                            0 = −1 ‫ إذن‬P ( 0 ) = −1 :             ‫و‬
                                                           . P ( x ) = x + 12 x − 1 :
                                                                          4
                                                                                                  ‫ود‬     ‫را‬        0              ‫و‬

                               Riyadiyate.site.voila.fr

				
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