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συμπληρωματικές προτάσεις αποδείξεις άλγεβρας α

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  • 2013-2014

    .

  • .

    ,

    ,

    .

    .

    2014

    1 1 , 5

  • 1

    2013-2014

    1

    = =

    x x x x , . = .

    1. 2. 3.

    1. x x 2. x x , x 3. ,x x x ,

    .

    1. = 2. = 3.

    1. x x x . x . x x . = .

    2. x x x , x x x =

    3. x x x . x , .

  • 2

    2013-2014

    De Morgan

    1. ( ) =

    2. ( ) =

    1. ( )x x x x

    x x x . ( ) x x x x x x

    ( )x . ( ) . ( ) = .

    2. ( )x x x x x

    x x . ( ) . x x x x x x

    ( )x . ( ) . ( ) = .

    ( ) ( ) ( ) =

    ( )x x x .: ( x x ) ( x x ) x x

    ( ) ( )x . ( ) ( ) ( ) ( ) ( )x x x , ( x x ) ( x x ) x x .

    ( )x , ( ) ( ) ( ) ( ) ( ) ( ) = .

  • 3

    2013-2014

    ( )0 1P

    : ( ) ( )0

    ( )

    ( )( )

    ( )( )

    0

    ( )0 1P

    ( ) ( )1P P = = (1)

    (1)

    x x x

    x x x

    =

    ( ) ( ) ( ) ( )P P P P = +

    ( ) x y = + , ( ) y z = +

    ( ) y =

    ( ) x y z = + +

    ( ) ( ) ( ) ( ) = +

    ( )( )

    ( )( )

    ( )( )

    ( )( )

    = +

    ( ) ( ) ( ) ( )P P P P = +

  • 4

    2013-2014

    ( ) ( )P P . (2)

    (2)

    { }1,2,3,4,5,6 = { }1,2 = { }1,3,5 =

    ( ) 13

    P = ( ) 12

    P = , ( ) ( )P P .

    ,

    x x x x

    x x x x

    ( ) ( ) = .

    ( ) ( ) 1P P + > ,

    . ( ) ( ) ( )P P P = + ( ) 1P > , ( )0 1P . ,

    .

    , ( ) ( ) 1P P +

    , ( ) ( ) ( )P P P = + .

    : ( )0 1P : ( ) ( ) 1P P +

  • 5

    2013-2014

    1. ( ) ( )P P 2. ( ) ( )P P

    1. ( ) ( )P P 2. ( ) ( )P P

  • 6

    2013-2014

    2

    , = :

    = 1 1 = , , 0

    ( ) ( )1 1 = = =

    , 0 1 1 1 1

    = = =

    , , , = = :

    = , , 0

    = 1 1 = = .

    1 1 =

    =

    , 0 = :

    0 = 0 =

    0 0 = : ( )1 1 0 =

    1 0 =

    1 0 =

    0 =

  • 7

    2013-2014

    , , 0 :

    = =

    ( )0 0 = = =0

    0 = = .

    , 0 :

    0 0

    0 = 0 0 = = , 0 . 0 = . 0 0

    , = :

    2 2 = = , *

    2 2 =

    = ==

  • 8

    2013-2014

    .

    .

    .

    = = = =

    =

    1. + = 2.

    = 3. ( ) =

    4.

    =

    5. ( ) = 6.

    =

    ,

    1.

    ++

    = = =

    2.

    = = =

    3. ( ) ( )( ) ( ) ( )( )

    = = =

    4.

    = = =

    5. ( ) ( ) ( )( ) ( )

    = = = =

    6. 0 , 1 = , :

    1

    1 1

    = = = = =

  • 9

    2013-2014

    1. 0 1< < < : > , *,

    2. 1 > < : < , *,

    1.

    0 10 1...0 1

    < < < <

    < <

    : 1 1

    < < >

    2.

    11

    .

    .

    .1

    > >

    >

    : 1 1

    > > <

    . =1 1.

    =-1 -1.

    -1,0,1. 4

    :

    1 <

  • 10

    2013-2014

    -1

    1 1>

    1

    11 0

    < < .

    1 0 < <

    -1 1 <

    1 1< .

    0 1< <

    , 1 1 <

    1 1> .

    1 > . , 1 1 >

    1 1< 1

    10 1

    < < .

  • 11

    2013-2014

    (*) ( ) .

    ( ) .

    .

    . [ ], [ ], . . x . x

    . x x .

    x

    , m ( = m

    , ). m x .

    x

    , ( = m

    , ). x M .

    m x M . m ,

    , .

    m

  • 12

    2013-2014

    [ ],y

    [ ],y . y .

    [ ], [ ], [ ],m M .

    .

    , ( ), 1,1 ( )1,1

    1 1 11 1

    1 1 1

    < < < <

    2. 2 2 2 + , ,

    3. 0 > , 1 2

    + 0 < , 1 2

    +

    1. , 0 > . :1 1 1 1

    < < < > .

    2. :

    ( )22 2 2 22 2 0 0, + +

    3. :

  • 13

    2013-2014

    ( )22 21 2 1 2 1 2 0 1 0,

    + + + .

    :

    0 = <

    0 < = = < .

    , :

    1. 0 + = +

    2. + , ,

    3. ,

    1. ( )22 + = + + = + ( )2 2 22 + = + +2 2 2 22 2 + + = + + 2 2 = = 0

    2. ( )22 + + ( )2 2 22 + +2 2 2 22 2 + + + 2 2

    , .

    + .

  • 14

    2013-2014

    3. 2 2 + + ( ) ( )2 2 +

    2 2 2 22 2 + + + 2 2

    , .

    + .

    , , :

    ( ) ( ) ( ), , ,d d d +

    ( ) ( ) ( ) ( ) ( ), , ,d d d = = + = + +

    , : 33 3 + = + 0 =

    3 x = 3 y = 0x 0y (1)

    : 33 3 + = + 3 33 x y x y + = + ( )33 3x y x y + = +3 3 3 2 2 33 3x y x x y xy y + = + + + ( )3 0xy x y + = 0x = 0y = 0x y+ =

    0x y= = (1). 0xy = 0 = .

    , , , : 2 2 + = + ;

    2 2 + = + 2 2 = ( ) 2 =

    : 2

    =

    2 ,

  • 15

    2013-2014

    = : 0 = = . = = .

    , : + < +

    + < + ( ) ( ) + < +( )( )( ) ( )...

    + < + + + +

    ( ) ( ) ( ) ( )1 1... + < + + + +

    ( ) ( )1 10 ... < + + 0, 0 > > .

    , :

    1. 1 1 < <

    1 1 > >

    1 1 = =

    2. 1 > : < >

    1 < : < <

    1. ( )1 1 1 < < <

    ( )1 1 1 > > >

    ( )1 1 1 = = =

    2. 1 > : ( ) ( ) < < < <

  • 16

    2013-2014

    1 < : ( ) ( ) < < < >

    :

    1. 1 < >

    2. 1 > <

    1. ( ) < < < 11 < 1 >

    2. ( ) > > > 11 > 1 <

  • 17

    2013-2014

    3

    ,

    x

    1 2,x x 0x + = 1 2x x .

    : 12

    00

    xx

    + = + =

    : ( )1 2 1 20 0 0x x x x = = = ,

    1 2 0x x . 10 0 0x + = = , 0 0x = x .

    2 0x x + + = , 0 1 2,x x , :

    1. 1 2x x

    =

    2. 2 21 2 4x x S P =

    1. 1 22

    2 2 2 2x x

    + + + +

    = = = =

    2. 2 22 2

    2 21 2 2 2 2 2

    4 4 4 4x x S P

    = = = = = =

  • 18

    2013-2014

    2 0x x + + = , 0 0 > , 0 < .

    2 2 20 0 4 0 4 4 00

    >

    < > > ><

    0 >

    0

    1 2S x x

    = + =

    2 0P <

    2

    000

    PS

    > > >

    1 2P x x

    = =

    2

    000

    PS

    = > >

    2

    000

    PS

    > >

  • 19

    2013-2014

    4

    ( )f x x = + , 0

    ( ), 0

    0 0, 0

    xf x x x

    x

    > >> + > > <

    0 =

    0 <