538
Lipschutz-Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e Front Matter Preface © The McGraw-Hill Companies, 2004 Preface Discrete mathematics, the study of ®nite systems, has become increasingly important as the computer age has advanced. The digital computer is basically a ®nite structure, and many of its properties can be understood and interpreted within the framework of ®nite mathematical systems. This book, in presenting the more essential material, may be used as a textbook for a formal course in discrete mathe- matics or as a supplement to all current texts. The ®rst three chapters cover the standard material on sets, relations, and func- tions and algorithms. Next come chapters on logic, vectors and matrices, counting, and probability. We than have three chapters on graph theory: graphs, directed graphs, and binary trees. Finally there are individual chapters on properties of the integers, algebraic systems, languages and machines, ordered sets and lattices, and Boolean algebra. The chapter on functions and algorithms includes a discussion of cardinality and countable sets, and complexity. The chapters on graph theory include discussions on planarity, traversability, minimal paths, and Warshall's and Human's algorithms. The chapter on languages and machines includes regular expressions, automata, and Turing machines and computable functions. We empha- size that the chapters have been written so that the order can be changed without diculty and without loss of continuity. This second edition of Discrete Mathmatics covers much more material and in greater depth than the ®rst edition. The topics of probability, regular expressions and regular sets, binary trees, cardinality, complexity, and Turing machines and compu- table functions did not appear in the ®rst edition or were only mentioned. This new material re¯ects the fact that discrete mathematics now is mainly a one-year course rather than a one-semester course. Each chapter begins with a clear statement of pertinent de®nition, principles, and theorems with illustrative and other descriptive material. This is followed by sets of solved and supplementary problems. The solved problems serve to illustrate and amplify the material, and also include proofs of theorems. The supplementary prob- lems furnish a complete review of the material in the chapter. More material has been included than can be covered in most ®rst courses. This has been done to make the book more ¯exible, to provide a more useful book of reference, and to stimulate further interest in the topics. Finally, we wish to thank the sta of the McGraw-Hill Schaum's Outline Series, especially Arthur Biderman and Maureen Walker, for their unfailing cooperation. SEYMOUR LIPSCHUTZ MARC LARS LIPSON v

Lipschutz−Lipson: Front Matter Preface © The …unina.stidue.net/Matematica Discreta/Materiale/Lipschutz, Lipson...Lipschutz−Lipson: Schaum’s Outline of Theory and Problems

Embed Size (px)

Citation preview

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    Front Matter Preface The McGrawHill Companies, 2004

    ! " # $%# &

    '

    (

    ) * (

    ! % *+$ ,# - , . " *

    ,)/*-01 234,5$06*.15 2.1, 234,-

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    ! " # $ %% && '" %% &&

    ' "

    ! ' '

    ( )

    * + , - )

    . " / %% " /&&

    " ! 0 %% && %%&&

    12 "

    . 3 "("

    12 $ " '

    4 5

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    " 6 (" 7 8 12 9 . 4 : 4 / ; '

    4 : 4 / " ' < ' 4 4

    1 2 9 . 4 : 4 / 4 4 4 5: (" !

    85

    0 -

    #$ ?" "

    % & '

    ; " ) "

    " $

    " ) " " )

    . " 4 : " '

    "

    ( )

    ; " $

    4

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    ;

    12

    : 7 > 4 : 8 4 8 ) " "

    4 : 5 8 7 >

    12 9 ! "

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    . /

    0+ $1 & $ $+

    F "

    @ "

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    4 %% && 0 ) :

    . /%

    " " %% && # #

    2

    $1

    0

    . @ %%&& G ) 12

    0

    . ) 12

    ; "

    " #

    . /(

    12 9 4 : : 5 8 4 : 8 4 : 5 8 : 4 : 8 4 :

    @!D E

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    12 9$ "

    $ $ (

    $ $

    *+ '".

    " D 48 ( '" " D :8

    ++

    A "

    0

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    .01$+$ 10

    4 !

    4

    $ 12 4

    142 # 1:2 " 1D 52

    .

    4

    :

    5

    8 7

    # ) 5

    . /2 . /3

    ++ 4

    % 0

    ( 1D 72

    ) 4 : 5 5 8 7 >

    4 : 8 7 > 4 : 8 7 >) 8

    3 -) . '

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    $ ; @ * 9

    @ $ " " "

    "

    " "

    ) ) ) 712 6 6 ) 712 ) 712 <

    '

    . /5

    7

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    0$

    *

    142 142 $

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    /

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    @!D E

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    < >

    !

    4 : .

    $*+$$ 10 9 "

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    9 0 4 : 12 : 412 " 12 :12 " : 40

    5 8 7 > / 12 " " 0

    4 5 7 / 4 12 " :0

    ;

    % .

    : > 4 4 : : 8 > 4 7 >

    ;

    .12 12 12 4 1 #2 4

    12 1 2 1 2 152 4

    12 " 12 12 : : 1 2

    12 4 > > 41 2 1 #2 4 4 4 4 152 4 4

    (

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    2 ).

    12 12 12 12 4 4 1 2 4 4 1 2 4 4

    A 1 2

    12 4 : 5 8 12 4 : 5 8 > 4 812 4 : 5 8 7 > 1 2 4 4 : 5 8 7 > 4 12 4 5 7 4 5 7 1 2 4 : 8 > 4 4 >

    (" 40 4 " 4 4 4

    3 ). 12 4 0 12 4 40 12 4 A .

    12 "

    142 J 1:2 J .

    12 5 8 7 >0 4 : 7 >0 4 4 5 7 0 : 8 > 412 4 :0 5 80 4 : 8 > 40 4 12 4 : 5 80 4 : 7 >0 4 5 7 >

    5 ). 12 0 12 012 40 1 2

    12 ) 4 5 8 7 4 12 : 4 5 8 7 >12 4 : * 4 :1 2 5 7

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    @!D E

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    5 6113

    6113

    7

    : -

    12 ) -

    12 ) )512 6 6113 "

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    12 ) -

    12 $ 6 56 6 6 6 6

    4 45 45 > 7 : 4

    . /2

    12 ' ) 512

    .

    : -

    : 7 6113 -> : 5 6113 -7 : -4 7 5 : 7 611345 7 : / 45 5 : 4 5/ 4 7 -

    12 7 / 4 :/ -

    -) . '

    , $ '.

    12 12 12 1 2

    ; '.12 12 12 1 2

    2 D" " . 12 12 12 12

    @!D E

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    3 D" . $+ $

    H" 4 :

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    12 * 1

    12 B 4

    12 *

    1 2 B

    % ) 4 :*

    4 :

    .

    12 . CE

    142 . C+, +4 :,E C+4, + :,E C+:, + 4,E

    1:2 . C+, +4, +:,E

    " J

    )

    ( D" " 4

    4 : 4

    4

    !

    4 : 4

    4 4 4 4

    "

    , D" 1 /.. 4 44 4: 4 4

    / 4 ! 4

    4 44 4 4 4 4

    44 44

    " /

    2 D". < "

    0

    ) 1 2 D " <

    @!D E

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    3 D" 4. '". 4 5 4 8 : 8 >

    4/

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    % ). 12 0 12 0 12

    %% ). 12 0 12

    %( ). 12 0 12 0 12

    %, 9 # # 5 4 # 5).

    12 1 2 4 1 #2 4 "

    12 12 4 152 4 3 12 4 1 2 4 1 2 4 4

    %2 9 D".

    12 # 12 #

    %3 D" .

    12 12 12 1 2 1 42

    %5 D" ! . 12 0 12

    %: J )

    & -

    (

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    ( = ) 5 3 5 1#2

    .

    12 12 12

    ( H # "

    -) . '

    (% $ '.

    12 12 12 1 2

    (( = " .

    12 12 12

    . > -

    (, H .

    12

    12 3

    12

    1 2 "

    12 '.

    48 457 8 8: / /1 2 1/ /2

    (2 = " 5. ;

    (3 ! " 4

    0 ! " .

    4

    >

    :

    ) . 12 0 12

    0 12 0 1 2

    0 12

    0

    1 2 0 # 0 5

    44

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    .

    (5 ) D 4 :

    (: ?" 12 4 5 7 4 4 : 8 5 7 >

    - & -

    ,2 = ".

    .

    4. *

    :. ; OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

    .

    1 9 9#0 !2

    @!D E

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    ,3 .

    . !

    4. F "

    :. * "

    H " . 12 A " 12F 12 ! 6

    ,5 D". 4 5 4 ,: D". 8 : 4 4:

    2

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    %% 12 50 4 5 12 5 80 4 : 5 8 >

    %( 12 : 5 > 12 : 7 > 12 : >

    %, 12 + ,0 12 + ,0 12 + ,0 1 2 + ,0 12 +,01 2 + ,0 1 #2 0 152 12 0 " + ,0 3 + ,0 + ,

    %:

    (

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    1. Set Theory Text The McGrawHill Companies, 2004

    (5 D 4 :4 . 4 : 4 : 4 : 4 4 : : 4 : 4 4 4 : 4 : : : : 4 4 : 4 : 4 : 4 :

    (: 12 12 )0 1

    2 )0 1

    2 0 1"2 0 1"2

    12 * :0 D 4: 7 .D

    , 4 7

    ,2 ) 4/

    # ; "

    . / 0

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    2. Relations Text The McGrawHill Companies, 2004

    ( )

    4 $

    $

    +

    5 0 5, 0 $

    * 0B *%4.&@5 ,(

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    2. Relations Text The McGrawHill Companies, 2004

    $

    ( ( = = =0 = = = (, ( ( = () = = = =

    $

    = = = ( = ( = ( = = = (

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    2. Relations Text The McGrawHill Companies, 2004

    !," D$ . !)" D$

    D$+ $ $ $

    1++$ % 1++$

    . $ $

    !" E % 06

    (

    ( 0 ( 0 ( ( ,

    ( , , , ( ,

    !" E % 07

    .

    , ) ),+ ( 0 ( 0 , ( 0 , ( 0 06 60. $ $

    !" E % 06

    0

    !( 0" !0 (" 0 ( 0 5 $ -

    .

    !" E % 07

    $ 5 1

    $ . $

    $ !@ $

    , , ,, , ,"

    $ 5

    + 2,

    $ ( , , ( 0 ,

    &

    ( ( 0 0

    . ! $ $ $

    !" E % 06 $

    , $ 0 ( ( , , 0 , , .

    $

    !" E % 07 $

    $ !" !"

    !"

    '2.> 0B *%4.&@5 ,,

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    2. Relations Text The McGrawHill Companies, 2004

    & $ 5 $ $ !9 # $ "

    $

    0 ( $

    '+ , . $ (

    3

    ' $ 4

    $ . 1 1

    1

    $ 1 9

    D$

    $ D$

    $

    F 2$ 5 1

    1 1 $ !> 0(6"

    1 1 2$

    1

    /0 % 1++$ 4

    D$

    2 '+ , 4

    !" D$ !" (

    D$

    $

    !" ' ( 0 , ) ( ( ( , 0 ) , ( , , ) ,

    D$ 0 0 ) )

    ) 0 , )

    ,) *%4.&@5 A'2.> 0

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    2. Relations Text The McGrawHill Companies, 2004

    !" ' G ! " $

    D$

    4

    4 * 0 ( 9

    (

    '+ !, $

    5 ' ?

    0

    $

    '+ (, 4

    $ 0

    $ . H ' ? 2 $

    ' ( 0 , ( 0 0 , , ,

    0 ( , 0 ,, , , 0 ( , 0 , , ,.

    $ 0 , ( 0 0 , , , ( ,

    5 67

    ' . ! D$

    $ #$

    !(" $ !0" !,"

    #$ # #$ +

    !(" $ !0" !,"

    & #$

    '2.> 0B *%4.&@5 ,-

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    2. Relations Text The McGrawHill Companies, 2004

    !" ' & % #$ #$ &

    !" #$

    !" #$ D$ $

    ! " 4 $

    ! " $ ) $ (( , ! )" ) $ (( , 00 6 ! )" ) $ 00 6 #$

    84 $ %

    #$ 1 *

    !(" % !0"

    $ !5 5 (C"5 #$

    +

    9 ! + ! #$

    #$ #$

    #

    '+ -, 4 #$ # 5

    !" $ !"

    !" '$ $ #$ #$

    $ > 00(

    !" ' ( 0 ,

    ( ( ( 0 0 ( 0 0 , ,

    ,6 *%4.&@5 A'2.> 0

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    2. Relations Text The McGrawHill Companies, 2004

    & D$

    $ #$ I

    ( ( 0 0 ( 0 , ,&$ ( 0 ( , & ( , 0 , $ #$

    !" 4 -

    - - J $ - - #$ $ #$ # -

    = (=- = - (= ( C) ( 6 (( 0 ?, 0 7 (0 , 70 , ? (, ) 6( ) C ()

    &$ # - = -

    #$ . #$

    = ( 0 , )I = ( 0 , ) 0( = ( 0 $ #$

    9 &

    . D$

    $ . > ' () $

    !"

    !(" !0" !,"

    !" D$

    $

    !" $ $ 2$ $ , , ,, , ,

    : .

    . $ / 1 1 ,

    '2.> 0B *%4.&@5 ,7

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    2. Relations Text The McGrawHill Companies, 2004

    !" 4 +

    !" # 0 0 0 ( & & # ,

    ( & = = =

    % ;+

    F$ ( 0 , !" + !" + !" !" 2

    ( ( 0 0 , , !" 2

    ( 0 , ( 0 ,!" 2

    . #

    F$ ( 0 , )

    1 ! 016" (0

    &$ 0 , 0 (0

    -

    ,? *%4.&@5 A'2.> 0

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    2. Relations Text The McGrawHill Companies, 2004

    4 ( 0 "9 $

    ( ( ( 0 0 0 ( ( 0 0

    2

    ( 0 5

    ( 0 &$ ! > 0)"

    ! >$

    ( $ 0 6 0 # # 2

    #

    0 6 0 $ , (

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    2. Relations Text The McGrawHill Companies, 2004

    ! " E *

    E ( , ) *

    3

    5 4 ( 0 , ) 6 $

    !@ J "!" 9

    !" E

    !" $ ( ' ( K

    !" $ ( 0 , ) 6

    (( (0 (, () (6 00 0) 06 ,, ,6 )) 662

    ( ( ( 0 ( , ( ) ( 6 0 0 0 ) 0 6 , , , 6 ) ) 6 6!" 5 01?

    !" *$ (

    ( ( ( 0 ( , ( ) ( 6 ( 0 0 ) 0 6 0 , , 6 , ) ) 6 6(

    5

    9 4 ( 0 , '

    $

    ( 0 0

    )= *%4.&@5 A'2.> 0

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    2. Relations Text The McGrawHill Companies, 2004

    !" !" $ $ $ $

    $ $$

    !" E 01C &$ ( ( + ( 5 0 0 9 $

    ( 0 0 !5 % 0-"

    9

    !" $ $ $

    $

    ( = ( =0 ( = (, = = =

    $

    = ( = ( = = = ( (

    $

    ( ( = =0 = ( (, = = =

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    2. Relations Text The McGrawHill Companies, 2004

    >$ 0( 4 % 5 & % & &

    9 & & $ $5 & &

    5 5 & $ & + & $ & & & 5 & & / $ & &

    .

    ' $ ( 0 , ( ( ( 0 ( , , , ( ( ( 0 0 ( 0 0 , , $ & ( ( ( 0 0 0 0 ,

    E $ !" D$+ !"

    +!" $+ ! "

    !" D$ 0 0 0 D$ , , & D$ D$

    !"

    ( 0 0 ( &

    !" & $ ( 0 0 , & ( , & $

    ! "

    ( 0 ( 0 0 ( 5

    F$ ( 0 , ) ' ( ( 0 0 0 , , 0 ) 0 ) )

    !" E

    !" * !" D$ !"

    !" $ !$"

    K

    !" 0

    !" 5 01(=

    !" !" D$ , , , , , !"

    )0 0 ) ) 0 0 )

    !" $ )0 0, ) , ) 0 0 , ) , !$"

    0, ,0 0 ,

    !" 5 0

    0 ( ( 0 0 0 , , 0 , , ) 0 ) , ) )

    )0 *%4.&@5 A'2.> 0

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    2. Relations Text The McGrawHill Companies, 2004

    :

    ! F$ ( 0 , $ !"

    !"

    !" $ ( $ $ &

    !" ( ( 0 0!" ( 0 0 ( 0 ,!" ( 0

    ( 5 &

    & >$ !" $

    &

    !" $ $ & $

    !" 5 & $ 5

    $ 2 & &

    !" 5 & $ 5

    $ $ 2 & & $

    - 4

    $ 1

    !(" 1

    !0" 1 1

    !" 5

    $ 1

    !" 5 1 1 1

    1 "

    !" $

    $ !(" / > 0(-

    $ !0"

    $

    1

    !" 4 & 1 " 5 1 & !(" & 1 !0" 5 & & 1 / 1 + & &

    '2.> 0B *%4.&@5 ),

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    2. Relations Text The McGrawHill Companies, 2004

    & 1 & . &

    3 '

    !" D$+ !"

    + !" $!" D$

    2

    D$ !"

    (

    2

    !" $

    0 , @ 0

    , 2

    $ 0 ,

    67

    5 ' ( 9

    $ 5 #$

    9 D$

    $

    !" $ ! " = $ 2 D$

    !" 5 ! " $ $ ! "

    !" @ ! " ! " $

    $ + ! " $

    . #$

    9 4 3 $

    >$ #$ 9 D$

    $

    !" '! 9 $ 2 D$

    )) *%4.&@5 A'2.> 0

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    2. Relations Text The McGrawHill Companies, 2004

    !" 5 .

    !" &! 5 0B *%4.&@5 )-

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    2. Relations Text The McGrawHill Companies, 2004

    * : ; :; : ;B

    &

    4 KL D$

    $

    $ !" + !" + !"

    ! ' E $ 5 !" D$+ !"

    !" $

    !" D$ (!" 5 !"

    ( !" ( !" ( ( ( ( ( ( ( ( ( 4 ( ( ! ("

    !" 5 2 $

    .

    4""+ 1 ;+

    ( 4 * :; + !% E$" !" * + + !" + * + !" + +

    - 4 & * ' & * & *

    3 !" 0 ) - 0 + !" 0 0 ( ( 0

    5 >$ !" + !"

    9 4 ( 0 , ) ( , ( ) , 0 , , , )

    !" $ ! " E

    !" !" !" (

    : 4

    !" + !" + !" + ! "

    4 $ # , (0+ , (0

    )6 *%4.&@5 A'2.> 0

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    2. Relations Text The McGrawHill Companies, 2004

    !" 9

    !" !" !" !" (

    !"

    % $

    !(" !," (=!0" # !)" ) (=

    E !" D$+ !"

    !"

    ! " $

    4 . $ ( 0 , !"

    !"

    !" D$ D$! " D$ D$!" $ $! "

    ! "

    (

    ! " D$ ( ! "

    (

    ! 5

    >$

    67

    ( >$ #$ ( #$

    - 4 ( 0 , (C 0= 4 #$ ! -" $ - #

    3 4 ( 0 , C

    !" >$ #$ !" 0 - #$ 0 -

    = 4""+ 1 ;+

    ( !" * + ! %" !> E$";!" + * !% " !E$ >";!" + + !% %" !% E$" !E$ %" !E$ E$";

    '2.> 0B *%4.&@5 )7

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    2. Relations Text The McGrawHill Companies, 2004

    - & * 01(( & * #

    3 !" ,+ 0+ !" 0 ,

    9 !" $ = = ( (= = = == ( ( (= = = =

    !" E ( , 0 , )!" ( , ( ) ( 0 , , , ) ,! " 5 01(0!" ( 0 ( , ( ) , 0 , , , )

    : !" !" !" ! "

    )? *%4.&@5 A'2.> 0

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    2. Relations Text The McGrawHill Companies, 2004

    !" C ( 6 0 , ,!" !" C 6 , !" ( 0 , !" ( C 0 6 , ,!" , ,

    !" @+ !" !0" !,"+ !" !(" !)"+ ! " !,"

    . !" ( 0 0 , ! " ( 0 0 (

    - ( 6 (( (6 0 7 (0 (7 , ? (, (? ) C () (C - (= (- 0=

    3 !" ( ) 0 - , 6 ) 7 - ? 6 C

    '2.> 0B *%4.&@5 )C

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    !

    " "

    # # $ % & $# % ! # &

    "! " ! '

    ( ( (

    & ) & &

    '

    $ %

    $% * + ( , ( ,

    $% - .

    " / $"% /

    $% " +01 )

    .

    23

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    4

    $ % -

    1 1 &

    1

    $%

    # )

    # )

    " # )

    5 # # # ) ! 6 4 ! # *! ! ) #

    6 # $ % !

    6 7

    6 ) ! !

    $% - # ) 8 +1$% #

    $% * 1 ( +#

    1 + ( + + 1 1 ( + 1 1 + ( 1 1 ( + 1

    )

    1 + ( + + 1 ( ) ( 6

    *.6/ +9 ":4*&4 64; 6-5&.7 21

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    1 ) 1 + 1 ( & + ( 1

    $%

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    6 # & # &

    6 # 1 &

    1

    ' 6 # 00 & # 00

    !

    00 00

    * 1# (# +# @# )

    " +0+ 4 1 00 ( 00 . + @ 00 + + @ @

    6 ( + ( + ( + 1 + 1 @ @

    1 00 + 00 @ 00

    . ( 00 00 . (

    1(

    ( ') * +

    ) 00 '

    # 00 1 ( A A

    6 00 A

    *.6/ +9 ":4*&4 64; 6-5&.7 2+

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    * #

    1 ( ( ( + + (( 2 = @ +

    " +0@ A

    1 A 1 1 00 ( 00 + 00 @ 00

    @

    1@

    +

    ,

    , -#.!-##& " !/0!#& # &(#.-

    0 '

    +

    - B )

    & 1 " +1@ +

    2

    ( ,2 C D D @ @+1@ @

    2

    + ,2 , D D @ @

    + #1 $

    - &4 $%

    &4+1@ + &4 2 ( &4,2 , &4D D &4 &4

    6< ) . 6< 3 3 3 6< 6<

    12 12 D D +++ +++ @@@ @@@ 33D2 33D2' 3

    2@ ":4*&4 64; 6-5&.7 ?*.6/ +

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    + 2 -+ #'

    - ! "

    ! $ "%

    $ ! "% ! " 7 ! $"% !

    ! "# 3 "' ! ! "

    (2 $ D% @ (2 $ 2% 3 +2 $ 11% ( + $ ,% +& ! ! " !$ "% " 3

    (= $ D% D 2 ( +D1 $ ,% , + 2 +C $ +% 3 )

    #

    $ "% " " $ "% "

    ! #

    3 " $ "% " $ "% "

    !

    3 1 ( " 1

    1 ( + "" 1(

    = C + D 2 11 1 2 , ( 13 3 1($ 3 " %

    !3

    ) $ %#

    % 3 1 1

    8 )

    "

    (@ 1= (@ 1(@

    11=

    1(2(+ 2( (2

    & )

    6

    )

    *.6/ +9 ":4*&4 64; 6-5&.7 22

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    &'

    - -

    ! 6

    ( , + (+ , 13 133 ( 13( 133( =@ = (= =@ 13 3331 + 13+ 3331

    "

    1 3 3 1 1 1

    3 )"! " 0

    13 +33 (@DD1 @3 +=,,C

    $. (D1,(,1

    # 13 (

    - (

    13 (

    "! ! B ( "

    ( 133 = (= =@ (D 1(,( 1333 C (, 21( (C 13(@

    ' 14 ' !3 + &'

    " +02

    ( 0 ( ( (

    " +02

    )

    2= ":4*&4 64; 6-5&.7 ?*.6/ +

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    & 7

    !

    5

    5 !6!!" !/! ! !

    !

    ' E '

    7

    6 # 1 ( + !

    1 ( + # ! 3 1 ( &

    3 1

    6 $ # 1 ( 1 (

    ) !

    $%

    !

    1 1( 1+

    1@ 1

    1(

    1@

    1,

    )

    1 (

    ) ! 1 ! 3$% ! 1 1 1 1 )

    11 ! 1

    ) ! 1 ! 3

    *.6/ +9 ":4*&4 64; 6-5&.7 2D

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    $% $% ) ) !

    1( A

    * 1+

    81"

    .

    $ 5 % * !1 ( +

    1 ( 1

    1

    !

    !

    1 11 ((

    2

    (( (( +( @( 2( @ C 1= (2 2@

    1 1 (

    8 += & 1(

    1 ( + 1(

    1 ( 23 2321(

    1(D2

    +3+ *

    - % 6 % # % " %

    # % % % % & 00 %

    )

    % # % % # %& % ) E ) & %

    1 ( 1 (

    2, ":4*&4 64; 6-5&.7 ?*.6/ +

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    - % 9 #

    # & ) " 1 ( 1 ( ( 1 .

    6

    : !$!&; !!

    6 $ ) & ) ) #

    $1% $(% 8

    6 $

    1

    F

    1 ( + ( 1& ) 3 1 )

    3 1 1 1 ( 1 ( ( + 1 ( + = @ 1 ( + @ (@

    2 1 ( + @ 2 1(3 = 1 ( + @ 2 = D(3

    2 2 @ 2 (@ 1(3 = = 2 = 1(3 D(3

    16

    ) #

    $ %#

    $% & 3 1$% & 3 1

    ) F 1 .#

    $1% F 3 $ 3 %$(% F )

    3

    6 ) 0)

    *.6/ +9 ":4*&4 64; 6-5&.7 2C

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    - @F ) ! #

    $1% @ @ +$(% + + ($+% ( ( 1$@% 1 1 3$2% 3 1$=% 1 1 1 1$D% ( ( 1 ($,% + + ( =$C% @ @ = (@

    #

    ) @F +F @F +F

    . +F ) (F +F (F

    ) (F 1F

    ) 1F 3F

    3F 3 )

    ' 3F ) 1F 1F ) (F (F ) +F ) +F ) @F

    &

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    =1 7>

    $% & 3 1 ( $% & 1 ( (( (1

    ) ) ((

    (1 .#

    $1% 3 1$(% ( )

    6 0)

    #

    6 3 1 ( ) #

    =# >

    $% & 3 1$% & 3 3 1 1$% & 3 3 1 1

    ) ) $% $% 3

    3 3 3 1 3 ( 3 + 3 6 )

    1 + / +(@ 8 ! 12 5 6

    & 0 )

    ? ##&;

    # 00 # 6 $ 1 ( 6 $

    ) "

    ) 9 6

    ' ' '

    ' ) 3 1 ( 1 (

    ) 3 $0% * 3

    *.6/ +9 ":4*&4 64; 6-5&.7 =1

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    $% ) + 1 + 2 D C 2$% - ( @ = # ) ( 00

    36 3 $ 6 )

    6 & $ / +12% !

    ' 6

    & 1(

    1 ( +

    6 ) / +1=

    ' % 3 1

    7 + + 1

    A !

    ) " ) # 00 '

    " % % # 3 1 # % ) 1 00 % ++

    * / +(,

    ' , => " 6 / $ / %

    !

    ' 5 ='+%>

    ' ! / +(C

    @ #&(.- #

    6 " ) 00 0) ) ' ' $. ' % "! ' " ' G ! "

    =( ":4*&4 64; 6-5&.7 ?*.6/ +

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    $/

    8%

    )

    (+ D( @ 12 2

    $% $ %# . 2

    2 (1(2 D(2 @2 D (23 1D2 (3 12 ,3

    + ( 1 = + &

    !

    1 1 1(

    $% $ %# .

    $ % #

    (( D @ 12 ( D @ 12

    2 +2 @2 12 1C2 12 C2 12 ,3"

    ! #

    2 ( D @ 1213 12 C2

    ( + 1C ,3 + + &

    . !

    * . $% G $%

    $5 * ;% - ) 5*; $% $ %# . ) ( (

    " (2, =3 #

    (2, # 1 ( + = ,= 1(C (2, =3 1 ( + @ 2 = 13 1( 12 (3 +3 =3

    6 5*;$(2, =3% =$% $ !%# . 1 $4 1 %

    1 ( $4 ( 1 4 1 ( + $4 + (% ' !1 !(

    1 ( + 3 1 5*; $ % " (2, =3#

    $1% ; (2, =3 1 1,$(% ; =3 1 1, ( =$+% ; 1 1, ( = + 3 ( = 5*;(2, =3

    *.6/ +9 ":4*&4 64; 6-5&.7 =+

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    8 G )

    5*; 11=

    A -0&!/; #&(.-

    E &

    G

    " A G " #

    & $% &

    H )

    " I ! A "! ! A 6

    A " 8 8J ) +0 * * * +0 * 8J * +0 A * 8J

    ! ) #

    $1% * # $(% # ! 1 ( ! 1 ( !

    11 (( !!

    & '

    ;66 ) &87 ' ) -* &87 ;66 -* 3 &87 ;66 &87 ;66 &87 ;66?19 ;66?(9 ) -* &87 ;66?-*9

    &87

    ;66?H9 '

    =@ ":4*&4 64; 6-5&.7 ?*.6/ +

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    $1% " ## * &87 ;66 &

    6 0

    $(% ## . &87 ;66 ! 6 1 ( + 1

    1 1 ( 1

    1

    1 ( 1

    1(

    1 1

    (

    ) &87 ! ;66

    A 7

    6 A

    * (4'2 % $

    " A * " &

    ( ( ( + (

    " +0= # (

    (

    :

    + )

    *.6/ +9 ":4*&4 64; 6-5&.7 =2

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    - ) '

    +

    ! !

    '

    + +$ + > %

    *

    & / +(D & +

    D( C @ +( ,+ 2D=( ,+( (@, ++

    38 * B4 #'

    6 ) +

    0

    #

    $% - # +$% < # + $%

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    *.6/ +9 ":4*&4 64; 6-5&.7 =D

    - ' 1 ( + @ ; ' ' $% ( + 1 @ ( 1+ ( @ @$% + 1 @ ( 1 1$% ( 1 + @ 1 @ ( 1 @ @

    ' ' # ' ' ' )

    $% 4 > $( +% $( 1% ( )

    $% 4 ( ' ) $% K 6 ( ' )

    !

    - ; )

    $% $%

    $% $ %

    6 #

    $% K

    $% K

    $% K

    $ % 4

    , #

    $% ( =$% + +( +

    )

    " +0,

    @

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    5 - # # ) " +0C " #

    A

    ' ) #

    4

    #

    : - ) ( 1 ( ( " )

    * # ( 1 ( 1( ( @( @ 1

    ( 1 ) ( (

    !#

    ) ( ( ( 1( ( @( @ 1

    !!" " # $!%&!

    ? ; 00

    $%

    $%

    $%

    $ %

    $% 4 7

    $% K

    $% 4 >

    $ % K ;> >

    @ - # # # ) " +013$% ;

    $% "

    =, ":4*&4 64; 6-5&.7 ?*.6/ +

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    C

    $% # + # ) #

    $% 4 ( @ 1 = ( @ . @ = @

    A * # # / #$% & 00 00$% & $% . 00

    " 00 6 00$% -

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    ##&;

    "

    $% ) $ % 13 (3 +3 @3 $% 1+ 2 11(, $% = D , C $% # ( 2$% (= (= 8 $% 2$% 3 ! 2 $ % 3 # ) 13 00 $% 3 # ) 2 00

    9 3

    00 9#

    1 ( + @ 2 = D , " " " " " " " "

    9 3 1 D1 ( D( + D+ @

    # 9 00 #

    ( 1 (

    6 9 3

    , - 1( ) / 8

    1 ( 1

    + 1 ( )

    " ) 1( )

    1 1 ! !#1 ( !1 E - 1 ( - # ) #

    1 ( 1 & ) & ) 00

    5 / +(# 6

    1(+ & 1 ( +

    ;) (+@ #

    ! # !" = 12 (@ ++ @( 21 ! )

    !!

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    : / ++# % 3 1

    % ) 1 1(1+ %

    00 # % - 1 1 ( ( % 1 ( + ' 1 ( #

    1 3111(1+1@ ( 3(1(((+(@ + 3+1+(+++@ @ 3@1@(@+@@

    3 1 ( C $" > 3(33 3333 31CC CCCC %

    - 31(+@ #

    1 1( 1

    4 %

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    $% (

    ( 1 1 ( + ( 1

    1 ( + ( 1 ( 1 ( + (

    (

    ( 1

    ( 1 ( + (

    8# $% ( , $% ( =@ $% 13 133 $ % 13 3331

    $% ( , + (+ , $% 13 133 ( 13( 133$% ( =@ = (= =@ $ % 13 3331 + 13+ 3331

    "! "

    13 +33 (@DD1 @3 +=,,C

    $. (D1,(,1

    !$!

    - , ) #

    , 3 , 1

    $% "# $% ,( 2 $

    % ,1( 2$% ' , L " ,$2,=1D%

    $% $% ,$( 2% M 3 (N2

    $

    % ,$1( 2% M ,$D 2% O 1M ?,$( 2% O 19 O 1 M ,$( 2% O (

    M 3 O ( M (

    $% 8 , 1 . ,$% ) ! ,$2,=1D% M ,+D

    - - ) #

    - 3 M1-( 1 1

    " -(2 " -(2 #

    -(2 -1( 1 $-= 1 1 -= ( $-+ 1 ( -+ + $-1 1 + -1 @ 3 @ @

    8 ( - 1 . -

    (- - ( 6

    , : ) 6 ) 1 +' 12 #

    D( ":4*&4 64; 6-5&.7 ?*.6/ +

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    $1% 1 + 31 ($(% 1 ( 31 1$+% 1 1 31 3$@% 1 3 3 1$2% 3 1 1 1 ($=% 1 3 ($D% 1 1 3 ($,% 3 ( ( 1 +$C% 1 1 +$13% 1 ( 3 +$11% 3 + + 1 @$1(% 1 ( @$1+% 1 + 3 @$1@% 3 @ @ 1 2$12% 1 + 2 ) $% )

    2 , 11 1@ $% @ $% 1 ( + &

    -!&! 0%&!-

    5 " 0

    #

    $% 1 ( $%

    (2 (

    $% ( + @ $ % ( 3 (' 0

    )$% ) ( 3 ( #($% ) $% ) (2 ( $2 2 # 2 2$ % . # 3 (

    : " 3 1 3 1 "#

    $% + D $% + (3 $% $ % $% 3 1+ 3 1D + D +$% 3 1(3 3 1+ + (3 (3$% - !

    . $ % - " "

    ? & 3 1 (( / /$%M- 3 3 1 1 % 1

    & 3 1 (( 3

    11

    3 1 "

    " 3 1 . & +" 2+ + ++ 2 @ 333 333( +2

    *.6/ +9 ":4*&4 64; 6-5&.7 D+

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    @ / +@ $*%# / $ / % # / ) 00 / & /

    / # / 00 - &

    # 4 # / &

    L & ) -

    / .

    A / ! 0

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    8 01

    C - * ; * * $% $% $% $ %

    - P '

    - * 1 ( + @ # * * ) " +01( $% ' $% " $% '

    - / 1 ( + @ 1 + ( 1 + @ @ + 1 ( ( + + 1 @ 1

    "# $% $% $%

    , - # ) + D " 1#

    00!!

    5 /# & # # 1 00

    : / +1# 6 # 00

    ? /# & # 1# 1 1 1 1

    @ " #

    3 1 # 3 1"# $% , $% # 0 $% # 1

    $% + D $ % # 0 $ % # 1 0 ) 2 1 )

    A * # % 3 % /#$% $% # % 3 # %

    *.6/ +9 ":4*&4 64; 6-5&.7 D2

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    ,C " #

    ( + @ $% "# $1% ( D $(% = , $+% + 1( $@% + 1($% / # 0 0 )

    ##& -%!

    , " #

    $% P7 Q

    $% #

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    3. Functions and Algorithms

    Text The McGrawHill Companies, 2004

    5C - , )

    , 2 , ( %

    " ,( D ,2 + ,12 (

    #4 8 01

    A $% 4 $% K $% 4

    C $% K $% 4 $% K $ % 4

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    4. Logic and Propositional Calculus

    Text The McGrawHill Companies, 2004

    ! "#

    $% % #& '() %% * + , +

    , * * -. ! % % -. * +

    / 0 * % ! 1 1+ 2! ! % )

    /0 3 + /*0 4 5

    /

    0 + /*0 6/

    0 7 /*

    0 , 8/*0 ' $+ /*

    0 %$+

    /*

    0 /*

    0+ *! /0! /

    0! /*0 ! % /

    0! /*0! /*0 +

    ! ! * * 1-+ ( +

    1 1 1$ % ! ! +

    /0 # * 1 % 1 # 9 1+

    /0 : * % 1 : ; : * +

    /0 1* /0 /*0 * < 1 1$ %

    +

    *

    1 * 1 % % %

    + *+

    =>

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    4. Logic and Propositional Calculus

    Text The McGrawHill Companies, 2004

    1 ?! ?! % ! *! % ! ! +

    !" % 1 1 1 %

    + (1!

    ! ? + ( *! * * + (.)

    #$ % ! < % + * 1 . -* 1 1 + 6;/0+ ! .

    % ! + ! + + "1* 1 1 % 1 + %1 +

    2 % )

    /0 3 6/

    0 3 @+/

    0 3 6/*0 3 @" . + ! 1 +

    &' (

    !" % 1 1 1 %

    + (1!

    ! ? + * * %+

    #$ % ! < % + * 1 . -* 1 1 + 6;/0+ "1*

    % 1 +

    23+ 6A '"B2 3#"3"("' 2'2&'&( =4

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    4. Logic and Propositional Calculus

    Text The McGrawHill Companies, 2004

    2 % )

    /0 3 6/

    0 3 @/

    0 3 6/*0 3 @" /*0 + 1 +

    #)*% % % %+ ( 1! ++! % * ! 1*! 1 1! ++! % * + ! % * C !

    + & % ! 1 + % 1 ) . 1 1 D +

    #')" B* ! ! ! 1 1 %

    1 ! 1! 1 % +(1!

    ! + * * %+

    #$ % ! < ! + * 1 . -* 1 1 + 6;/0+ *

    % * +

    2 % )

    3 + @ 3 + @7 3 + 7 @

    7 < 7 + ( ! 7 < ! 7 +

    #)*% * ! ! ;E+ ! )

    +

    ' *1 ! ! ! % $ * #& /0 '( /0! * ! ! / 1;-0+ ( % 1

    >F '"B2 3#"3"("' 2'2&'&( G23+ 6

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    4. Logic and Propositional Calculus

    Text The McGrawHill Companies, 2004

    * * * *1! ! * $% * *1 $%+ % % + ,1 % 1 1 1%+

    2! ! + 6; /0 % 1 + "1* . 1 *1 % 1 % 1 1 + / *1! 1*! 6 % < 7 *1! > % < ! ! *1! % -+0 ! * 1 * 1 . * + % 1 * ! % +

    1 % + 6; /0+ + 6; /0 % *1 < % 1+

    &' (

    #)*% * * 1 ! % *+ (.!

    * % * ! +

    ,#)-# #. / !' ) . )0,#

    % 1 %)/0 % 1 % + 6;7+ ! . % *1

    1 *+ % % *1 % H *1 * + . % 1 (+

    &' (

    23+ 6A '"B2 3#"3"("' 2'2&'&( >

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    4. Logic and Propositional Calculus

    Text The McGrawHill Companies, 2004

    /0 ! * 1 * % + 6;6+ ! . * *1 *1 ! * + , % * 1+

    1 *1 ! ! 1+

    &' (

    1

    ( 1 ! %! * *1+ ( +!

    1 ! %! * *1+ ! ! ! ! ! ! ! ! + *. 1 $ 1 + 6;@+ / 1

    * % % *1 +0

    &' (1

    % ! % +

    % 1 ! 1 + ( * *1 % 1 ! * + %)

    > '"B2 3#"3"("' 2'2&'&( G23+ 6

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    4. Logic and Propositional Calculus

    Text The McGrawHill Companies, 2004

    .## 2!,# / 03% !

    4 56

    % 1 ! ! 1

    * 1+ 2! ! 1 + 6;5+ "1* 1 1 ! ! 1 . + ! % %

    %! -*+

    #)*% 2

    * 1

    1 % %)

    * 1

    %*! 1*! +

    # ! * I+

    * +

    &' (4

    7 &

    3 * % % 1 6;+ / 1! * ! *+0 , +

    .## % 3 % 1 6;+

    23+ 6A '"B2 3#"3"("' 2'2&'&( >7

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    4. Logic and Propositional Calculus

    Text The McGrawHill Companies, 2004

    8

    ! ! + ( 1

    - +

    + ( 1

    &' (7 &' (8

    >6 '"B2 3#"3"("' 2'2&'&( G23+ 6

    )0,# ( )9 / .# ),'#0) /

    ## ,)9

    /0 /0 !)-# ,)9

    / 0 / 0 )-# ,)9

    /70 /70 0-# ,)9

    /60 /60 #: ,)9

    /@0 /@0 /50 /50

    ,## ,)9

    /=0 /=0 />0 />0

    -, ,)9

    /40 #'); ,)9

    /F0 /F0

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    4. Logic and Propositional Calculus

    Text The McGrawHill Companies, 2004

    * . 1 1 + 6;=+ "1* )/0 % . +

    ! % ! * /0 1 %* * * %+

    1 + 6;>+ "1* 1 ! ! 1 + ! -* < !

    %! -* % ** * % + , 11* ; +

    <

    * ! ! / -0 ! + ( 1

    * E %)

    #$ % 1 %* +

    % *

    /0 % *)

    ! / 0 % 1 + 6;4+ (.! 2 /%0 ! +

    /0 % )

    1 2 /%0 7 1 + 6;4! 1 +

    &' (@

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    4. Logic and Propositional Calculus

    Text The McGrawHill Companies, 2004

    .## % * +

    , +

    )

    ! +

    ! % *)

    / 0 *. 1 1 + 6;F % % % )

    -*! * 2 /%0! @! = >! + /"1* 1 - 7 > *1 ! +0

    &' (=

    , % 1* ** . + , E * * ! 1 + % +

    2 % )

    ) 1! +

    ) ! +

    +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

    ) I +

    1% ! 1*

    + , *+

    % 1! < 1 6+5 /'% (0 *+

    >5 '"B2 3#"3"("' 2'2&'&( G23+ 6

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    4. Logic and Propositional Calculus

    Text The McGrawHill Companies, 2004

    =

    ! %

    %* +

    , + 1 + 6;+ "1* 2 /%0 ! + +

    &' (

    % %* !

    *< *+ ! * ; % ! ++! + , +.## % ! %

    -*)/0 +/

    0 *+/

    0 +

    , % % ! 1% + + + + % ! ! 1* +

    &" 5&

    ' 1 * + /) 0 .

    % + ! 1 /% *0 %* 1 *1 + ! % + %!

    ) J ) -! % 1! - -

    ** *1 +

    23+ 6A '"B2 3#"3"("' 2'2&'&( >=

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    4. Logic and Propositional Calculus

    Text The McGrawHill Companies, 2004

    . * +/0 ' 1 =+

    ) = 5 = > @+

    /0 ' 1 @ 7+ ) @ 7

    + %! * +/0 ' 1 @ +

    ) @ * +

    #)*% 1* % . 1 ! ! + % 1 -. +

    -#), 5)$#

    ' 1 . + 2 !

    % * ! ! ! ! + 1

    % * "+ / !0 -*

    # ! +

    1 *+%*! ! 1 -. ! * * % % -* / !0 / #0+ (.)

    ) ) ! < %! +

    /0 6 7 ) 6 7 7

    /0 > ) > = >

    /0 1 1 . ) $ %)) $ ) $

    >> '"B2 3#"3"("' 2'2&'&( G23+ 6

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    4. Logic and Propositional Calculus

    Text The McGrawHill Companies, 2004

    >#), 5)$#

    ' 1 . + 2 %

    % ! ! ! + 1

    % "+ (; / %0 -*

    ++! + ! ! ! 1 -.! * *+ (.)

    ) ) < %! +

    /0 6 = ) 6 = /0 5 6 ) 5 6 +/0 1 1 . ) $ %)

    ) $ ) $

    )

    ' 7 @ 1 1 ! + % *

    1 1

    7 @ % -*

    * 1 (!

    % * +

    1 1

    7 @ % -*

    1 %

    1

    #)*% % . ! /0 1 % < 1 /0 % 1 ! * % .+

    23+ 6A '"B2 3#"3"("' 2'2&'&( >4

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    4. Logic and Propositional Calculus

    Text The McGrawHill Companies, 2004

    & 5&

    2 ) ? + ) ? ! -*! ? % / 0

    (1! ?! 1* 1 %

    & 0 & / 0! % !

    & & 1* /0+ )

    .## 1 2#')3% %! % % -*)

    /0 ! ! +/ 0 +

    % -.+

    .## 4 2#')3% ! % % -*)

    /0 ! +/ 0 ! +

    /0 % * )

    * % * > * >

    /0 % * )

    /*0 % @F

    * * @F

    #)*% 1* < ) % ! * *

    3*! % < + (! ! ! . 1)

    % (! ! ! . 1)

    % )

    /0 /

    0 /

    0

    4F '"B2 3#"3"("' 2'2&'&( G23+ 6

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    4. Logic and Propositional Calculus

    Text The McGrawHill Companies, 2004

    " % % + ! %

    * %),#

    6+5 % ! -* % ! %! F % F + ( F +

    /0 2 F+ F ! ! F F +

    /0 2 ! + ! ! +(.! ! !

    /0 2 + % * + %! * % +

    ), &! 9. # .) # 6))0,#

    / *10 . * ;

    % ; + !

    7' >

    7 + ( *+%*! % * %)

    )! !,#% 1 -. *1! !

    ' '

    *+

    ' ( 7 4 F+ ; ( ( (/0 % -. *1)

    ! * ! F + ! ! 4< ! >! +

    /0 % )

    ! ! ! * ! % * F < +

    K 1% /0 /0 -.+ K -. K + , ! % -. ! - % +

    23+ 6A '"B2 3#"3"("' 2'2&'&( 4

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    4. Logic and Propositional Calculus

    Text The McGrawHill Companies, 2004

    #')' 5)$# )## 9. # .) # 6))0,#

    L. % *1 1 1 * 6+@ 6+5+ 1 + !

    ' ' ' ' ' ' ' '

    ! % % -. +

    /0 2 -. )* %

    ) *

    /0 . ) - %)

    F F F ) ) - %)

    F F F )

    ,-# 0,#

    ' 1 1 + B* *1 % 1 % ) /0 < /0 < /0 < / 0

    ! ! ! ! ! ! *! +

    /0 + /0 +

    /0 + / 0 +

    ' 1 $ *

    +! 1 $ * ,+! 1 $ + , % 1 )

    /0 $ *

    + * ,+! 1

    /0 $ *

    + * ,+! *

    +

    /0 $ *

    + 1

    / 0 $ * ,+ 1 *

    +

    & ! /! -*! 10! /0+

    /0 < /0 < /0 0< / 0 +

    4 '"B2 3#"3"("' 2'2&'&( G23+ 6

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    4. Logic and Propositional Calculus

    Text The McGrawHill Companies, 2004

    + 6 +

    * % )

    /0 6 @ 5 7 4 /0 6 @ 4 6/0 7 @ 5 = / 0 7 @ 6 =

    % 1 1 + ) /0 < /0 < /0 8-?" 7@ A7?48>" $+%

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    =

    A '$ "'&

    " 6 4 6 /

    #

    +$

    $%

    $)

    (

    $&%&$:

    /

    9

    '%

    &

    %$)

    ) % $+(

    ,

    :%(

    $:

    )

    $' % , )+ )' : $( '+

    '

    )

    !

    7

    $$ $) $)$ )) )

    $ )

    6 3

    7 6

    7 7 6 +

    $+& =>8-?" 7@ A7?48>" B8 7; '

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    $ & '+ % )

    ) % 4 $& ' + %)

    $+

    &%

    ')

    ) & 6 + + + + ++ + + +

    "

    )

    % *

    $ '

    % $ ) * '

    ) $ & $

    " !# ! ! !$! $%$!

    9 6

    $$ $$ $) $) $ $)$ )$ )) )) ) )

    $ $ ) )

    $$ $) $)$ )) )

    $ )

    - 3 / $

    . 6 /

    9 $ ) %+ & '

    & ( ,

    $ % *

    $ & ) ( % ,+ $ & % ' *

    ' & $$

    $ $ )

    % %$ %) %%%+ %& %'

    % ( :

    + $) $'

    ) % ) & (+ , $+

    $) $, )&% : )$

    $+ )) $,% $* %$

    8 7; '< =>8-?" 7@ A7?48>" $+'

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    A

    & ' 9 6 #

    + + + $

    @ / + 6 7

    $ )

    ) %

    9 / $ $ '$

    '$ " ; '$+

    !# $%$!

    / " %' C2

    / #

    $ )

    $)

    $$ ))

    $

    3 6 $ $ / .

    *& '

    %)

    $

    *% &) '% )$ , ' ,

    ($ , %

    &

    :)'

    )& : $( $' %)

    3 /

    $+( =>8-?" 7@ A7?48>" B8 7; '

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    8 7; '< =>8-?" 7@ A7?48>" $+*

    (' " 0

    $$ $ $ $

    $$ $ $ $

    $$ $ $

    $$ ))

    $

    3 6 /

    $ %) $

    ) + &

    ' ) (

    " ) ) ) % / ) % / / $ %

    )'

    +)

    &(

    ) $' + ( & $,

    $* ( $&

    )$

    $* ( $&& ' + ) , (

    $* ( $&$ ) $&

    " $ )% &

    ' (

    + )

    ' + ( &$' + $, ,

    ' )

    $' $+

    ' $, $+ )&

    + ( + ,

    )% %&( ,

    > '(

    A #

    & ' 9 /#

    3 + + + + + 6

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    ) * + , $ -

    7

    D 2

    ) % &' ( , :

    $ ) %' ( ,

    &:

    -

    ! $ ) % &' ( , :

    . !%

    "

    $ ) %& ' (

    "

    $ &) '% (

    $%'" $

    %'

    @ " 4

    /

    & ' 9 /#

    " " " " "" " " ""

    3 6

    / 0! !

    7 7

    $$ ))

    # # 6 # $ "/

    # #

    $+, =>8-?" 7@ A7?48>" B8 7; '

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    8

    $ ) ++ & (' % )

    $ + + ++ $ + ++ + $ ++ + + $

    5 / % $ & ) &0 6 &

    !1 , -

    9 4 #

    ) % ) $ + # +; / "/

    $ + $ ))

    / $ $ +# $ ))

    @ $ $ +# + 4 $ 6

    $

    " $ )% &

    ) $ )% &

    $ )% &

    * (: ))

    % ) * (: ))

    $ )% &

    $$ %,

    '* $+(

    " $ )) % '

    $ ) * (: ))

    % $ )% &

    ' $ +

    + $

    $( $,)* ($

    " ) % $+

    * (: ))

    % $ )% &

    $+ $ +

    + $

    + +

    + +

    6

    2 3$ 45$!6 !7

    7

    #

    # " ; ')&0 $ -

    ) '$ %

    % '$ )

    8 7; '< =>8-?" 7@ A7?48>" $+:

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    ( ' $+ $+% % ' (

    $ +

    + $

    ( ' $' $') ) ' (

    $ +

    + $

    4 2 # # 0

    $ + )) $ %& $ ,

    $$ ) )& + $( $ $

    $$ + $) ) + ) ) + ))) & $, & + % & $ %&& & &, , + , , $ ,

    $ + ++ $ ++ + $

    8 !

    /

    $$ $) $)$ )) )

    $ )

    3 6 $

    $ ) % / #

    $$ $$$$ $)

    )$ ))

    $$)) $))$$$ $) $%

    )$ )) )%

    %$ %) %%

    $$))%% $))%%$ $%)$%) $%))%$ $))$%% $$)%%)

    )#

    $$+ =>8-?" 7@ A7?48>" B8 7; '

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    % #

    3 6 2

    ' &) %

    '% &) $' , *) $

    & (

    )( $& $) & $(

    ) $ %

    & ( $' $ +

    )(+ $$' %$& %(' $&+ )$$

    + ' $) :+ + ) ,$

    5 ( ,

    / #

    $$))

    2 $ ) $ ) $ $ 3 / % ; 8 (

    7 #

    & "' 9

    * 9 , 9 ) )

    3 $ "/ 2 )) & $ $ ) )

    $ )$ )

    $ +

    + $

    $ $ ) )$ $ ) )

    $ +

    + $

    8 7; '< =>8-?" 7@ A7?48>" $$$

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    " ) ) #

    $ $ $$ $ +

    ) ) +) ) $

    - #

    $ +

    + $

    @ D " + 2

    $ $ ) )

    $

    $

    )

    )

    7

    $

    $

    $

    4 # 3 + ) ) #$ 4 ) 2 % A $

    ) %& '

    )

    $ $)' %

    & )

    ')

    ,)

    ) $

    - + 2 $ $ ) ) $ 7 @

    & ' 7 6

    : $ !; < %!7 5!! $ ! 4%!$6

    C

    + =

    8 %$ %) % /6 % 6 7 6

    6 6

    #

    $$) =>8-?" 7@ A7?48>" B8 7; '

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    8 7; '< =>8-?" 7@ A7?48>" $$%

    $ 4 % % # EE4 % %FF

    ) A % 6 # EEA % FF

    % 7 % % % % % # EE7 % %FF

    B< B< 0 #

    B< 7 % 6 % % % % + 3 # EE7 % %FF

    3 6 B< B

    8 / &

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    3 6 6

    $$ ))

    6

    ? ' " $

    $$$

    ? ' - " $

    EE7 $%$ $$%FF

    $ ) % $ )) & & ( $+% ( ( : $%

    $$& =>8-?" 7@ A7?48>" B8 7; '

    !1& :! >=* # / 6 4 >G4 3

    6 - $ 7 $$ + 6

    / $ 1 $$ 6 $$ $#$ " $$$ ) 7 '$ '

    B % $$$%$ %

    ? " $ /

    ) / 6 " ) )) +

    8 6

    !1& :3 3?=* # $$ ))

    A 6 % $ $ 1 $ 6 $ ) $#

    $ " ) 7 % %4

    EE7 $% %FFB % % %

    ? " $ %$ %) %) A %$ $$$

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    7 $+'7 "/ $$ $ 6 $$ EE7 )%$ %)FF EE7 %%$ %% )% ) + )% EE7 %)%) %%FF

    $ ) % $ )+ + ) & (+ + % ( *

    $ ) % $ )+ + ) & (+ + + + )

    @ 7 $+'5 "/ %% $)

    %' $ %' $ 6 EE7 (%% %)FF EE7 )%% %$FF #

    $ ) % $ )+ + ) & (+ + + + $

    $ ) % $ ++ + ) & ++ + + + $

    A %) $) )% $ )% $ +

    EE7 %%$ %$FF #

    $ ) % $ ++ + $ ) ++ + + + $

    $ ) + * ++ + $ ) ++ + + + $

    7 $+'7 $+'5 7 #

    & .' 7

    $

    ) , + , $ -

    8 ! C !

    ( %

    # ? ! 4 + + + +

    (

    # !

    !

    / ; '%) '%$

    " #

    ) %) ' &% ) '

    8 7; '< =>8-?" 7@ A7?48>" $$'

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    ? ! #

    ! $ ) $ %

    ) ' $ &% ) $ '

    $ ) $ %

    + $ % $++ , & &

    $ ) $ %

    + $ % $++ + ), ,&

    $ ) $ %+ $ % $++ + $ %

    $ ) + ++ $ + $+ + $ %

    $ + + )+ $ + $+ + $ %

    ) $ % )$ % 3 !

    9 , )8 % % $ /

    % % #$$ $) $% $)$ )) )% +%$ %) %% +

    $$ $) $% +)$ )) )% $%$ %) %% +

    $$ $) $% +)$ )) )% +%$ %) %% $

    @ D 7 #

    $ + )) $ %& $ ,

    ! # ! #

    ! $ + ) $ + +) $ % + $ +& $ , + + $

    $ + ) $ + ++ $ $ ) $ ++ $ + & + $

    $ + ) $ + ++ $ $ ) $ ++ + $ ( $ $

    4 ! 0 !

    #

    ! $ + + $$ ) )+ $ + & + $+ + $ ( $ $

    $ + + $$ ) )+ $ + & + $+ + $ ( $ $

    $$( =>8-?" 7@ A7?48>" B8 7; '

    !1& :' ) ! # 0 !

    # !

    ? ! 4 6 ! -

    !

    ! #

    # !

    " $ !

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    / 0 $ 4

    $ $$ ) )& + $( $ $

    3$! 4@A6 !

    + $ 8 #

    + $ + $+ + $ + + +$ $ $ $ + $

    = + 79"> $ ?1> -? 7@ 0

    + $

    5 8 $'

    @ + $ ! 5

    5

    $ $$ +

    + $

    + $

    + + $ $

    + + $ +

    + $+ $

    - 5 5 / 6 $

    9* %

    9 )* $ % + & + ', # 0 0 %0

    7 #

    )* $ % + & ) %* + $ & $* ' 7 #

    % + & + ', % + + ' & , % '& A %#

    % %)* $ ( )$% 8 $0

    + ', +' ,

    8 7; '< =>8-?" 7@ A7?48>" $$*

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    9 ; '$ # % &0 ) % '

    % & %)* $ &% + & ()$ % $) +$( $,)$$% ) % (

    ))* $ %% + & '+ ', &$& ) : + $) +)' &+ '%: '&

    9 ; '$ # 0 0 A #

    )% *+ $& ( + & ) )+ *' $, + %' , &% %+ +' &, + + %) %)

    " # %$)&0 )% ,* / )

    )

    ) %) $)) &) : $&& $( $(: $(: $%

    ) & : (& &: $)( $)(

    $ $ )$ $ & #

    $ $ )$ ) ) $ & 0 ) $ & % $

    . " '%

    &

    $')

    %$)

    # ' )0 ) & %

    ' ) ''%

    &

    )$')

    )'$'

    )+

    )

    $+&

    )*'

    )&

    ) & % $+(,

    &)+,

    :%(

    )%$*))

    !# ! ! !$! $%$!

    / C $ ) %& ' (

    $ $ )

    + % '

    # 0 % &

    7 #

    $ $ ) $ % )& + ' % ( '

    ) $ $

    & ) $

    $$, =>8-?" 7@ A7?48>" B8 7; '

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    A #

    % %$ %) %%%& %' %(

    % ( :

    $) $' $,

    & &$ &$ &)&+ &% &'

    & & ,+ $) )+

    2 ) % $ ) %& ' (

    % + )* $ ,

    #

    ) % ) & (, $+ $)

    : + (

    )$ % )&

    * & +): * %(

    @ % %

    8 %

    ($ )

    &

    %

    #

    % %% %

    & (

    $ ) %

    "

    % &% (% $% ) %

    ) &) ( ) $ %

    ) & $ %

    : ; '$# 9

    -

    !# $%$!

    8# %) '

    ($

    &

    )$ * &

    '%(:

    A #

    %) '

    ($

    &

    %( )$ '& $, ) )+ &

    )$ * &

    '

    %(:

    $+ % &) %( *

    8 7; '< =>8-?" 7@ A7?48>" $$:

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    $)+ =>8-?" 7@ A7?48>" B8 7; '

    9 6 /# ) %% & $ )% $ & &% % & $$ ) ' )) % $ ) )) &

    4 / 6

    #

    ) & @ / @ / & ) ' % $ ) &

    9 $ %) $

    ) + &

    % ) (

    # 0

    " ) ) ) % / ) % /

    / $ % )%

    +)

    &(

    #

    $ %

    ) $

    ) + &% ) (

    $) %% $+ %$ $& %(

    ) : + ( & $,

    $$ ( $&

    )$ #

    $ %

    ) $

    ) + &% ) (

    $$ ( $&

    & % + ) , (

    $$ ( $&$ ) $&

    @ ) % ) ) " % ) /

    " 8# $ (

    % '

    & +) $

    $ (% '

    )

    *

    0 $

    (

    $ (% '

    0

    $(

    % )0 )$ $(

    / ) ) ) ) / ) ) #$ (

    % '

    & +) $

    & $) + $)$) $+ + '

    $( () '

    / ) ) ) $ / ) $ #$ (

    % '

    )*

    ) &)( %'

    &+&$

    @ / ) $ ) ) " $ ) . /

    / ) $ $ ) / ) ) #$(

    % ) $% $)

    (% ()

    % )$, $)

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    / $ ) ) $ / $ $ #

    )$ $(

    )$ $( ) ( ,

    ; ')# 9 "

    $$ ))

    $

    $ ))

    @

    $$ ))

    $

    $

    $

    - "

    $$ ))

    $

    $

    $

    "

    !%

    . #

    $ ) %* , :

    $ ) %) & '% ' (

    $% '* )

    )&(

    ? #

    " $ *

    ) ,% :

    "

    $ ) %) & '% ' (

    "

    $%'

    *

    )

    " )& (

    @ " 0 @ )

    / 9 $ ) +% $ &

    " 0 "

    " # " $ %) $+ &

    " $ ) +% $ &

    $ %) $+ &

    $ & + % ) +

    % ) + : $ $(

    ' $$ )(

    8 7; '< =>8-?" 7@ A7?48>" $)$

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    " $ %) $+ &

    $ ) +% $ &

    $ : ) % + $)) % & $ + &+ $) + & + $(

    $+ $ $)$ ' &$) & $(

    2 ; '%# " "" "

    $$ )) *

    * B < " - " "

    8 ""

    $ )

    $)

    $ ))

    " ""

    0! !

    8 #

    $ % () ' ,& ) *

    ) %& '

    0 $ ) %

    & ' (

    $$ )) %% $ ' * ) ' /

    : 9 $ )& %

    # )0 %

    ) $ )& %

    $ )& %

    $ , ) (

    & $) , :

    : &, $*

    % ) $ )& %

    : &

    , $*

    : $( & %&%( )& $( '$

    * %+

    (+ (*

    9 $ )% & ' ) ) $$ ; ')+ /# $ 0 8 $ / '# ' $ )% & '#

    $ )% & '# ) * %+(+ (*

    & $ )

    & %

    ' $ ++ $

    $)) =>8-?" 7@ A7?48>" B8 7; '

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    #

    $ $& (+$)+ $%&

    & ,$( $)

    ' ++ '

    9 #

    $ $& & ' (+ , +$)+ $( + $%& $) '

    $% ')

    $+& $$*

    8 / $$# $$ ) ) $$#

    ) ) $$# : &, $*

    ) $ )& %

    $$ $ +

    + $

    : &, $*

    ) &, (

    $$ +

    + $$

    + ++ +

    " + 6

    ! !

    8 #

    & '

    % )

    0 ) *+ (

    0

    0

    1

    #

    & '

    % )

    &) '% , $' *

    ) *+ (

    )( *+ $) + $)

    ) ) ) ) ) ))

    ) ) ) ) )

    #

    $ ) %& ) %+ ' $

    0 & $ )+ ) %' ) $

    0 ) % &$ ) %

    $ ) '

    +# 1 " ':

    $ ) %& ) %+ ' $

    ) + (+ + $' , ''

    & $ )+ ) %' ) $

    , $' + )+ )& + (*

    ) % &$ ) %

    $ ) '

    )+ : , , $) $' $&

    8 7; '< =>8-?" 7@ A7?48>" $)%

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    " #

    ' %& )

    0 ) %

    $ %

    0 ) (

    % :

    ) ) " ': / ') %& $+ $) ) @ 2

    $#

    $ $)

    ) %& '

    $

    %)

    ) ')

    / )% %$ ( % : @ 2 $#

    $ $:% %

    $ )

    $%

    $%

    $: ):

    / ): (% $, $, + " +

    $ ) )) % ($ $ *

    ! # ! #

    ! $ ) ) $ + +) % ( + $ +$ $ * + + $

    $ ) ) $ + ++ $ ) ) $ ++ % ' $ + $

    $ ) ) $ + ++ $ ) ) $ ++ + $ ' % $

    4 ! 0 ! #

    ! $ ) + $$ ( )+ $ + , ' )+ + $ ' % $

    $ + + )* $( $+ $ + , ' )+ + $ ' % $

    / # $8-?" 7@ A7?48>" B8 7; '

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    / 9 4 # "

    9 $ ) 4 $ $ # ) ) # $ $# $) $) #) )

    2 " " $ $$

    3

    $$ $$ #$ $ #

    $$ $$ $# $ # $$ 0 $ $$

    B$ !7 < 7 5!! $ !

    8 4 #

    + $ % & (& + ) ' %+ + * ) ,

    0 + + + + +$ ) % & '+ + ' & *

    0 + ) ) ) )+ % $ + ++ + + + +

    4 /

    5 6

    @

    : ? #

    $ ) % +) & ) )% ( & %

    1 $$ 6 $$0 EE7 )%$ %)FF EE7 %%$ %%FF

    $ ) % ++ + & )+ + ' %

    @ )% & 6 )%0 EE7 '%) &%%FF

    $ ) % ++ + & )+ + + )

    8 7; '< =>8-?" 7@ A7?48>" $)'

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    3 H

    $ ) % + $+ + ' ) &+ + + * %

    + $ * ' ++ + + + $+ + + + +

    $ + ' + )+ $ ) + &+ + + $ *

    / 6 ' *

    $ 7 6 6 ' *

    ? #

    $ ) % $ )$ $ & $ %) ' : ) ,

    EE7 %$ %)FF EE7 )%$ %%FF EE7 %%) %%FF

    $ ) % $ )+ % $ ) $+ : % & &

    $ ) % $ )+ % $ ) $+ + + ) $

    @ 2 "/ / %%

    $) %& $ EE7 )%% %)FF EE7 %%

    %$FF

    $ ) % $ )+ % $ ) $+ + + $ $)

    $ ) % + %)+ % $ + )+ + + $ $)

    @ %) $% 2 )) $ EE7 )%) %$FF 3

    $ ) % + %)+ $ $% +

    )%

    + + + $ $)

    $ + $$% +

    $*(

    + $ $% +)%

    + + + $ $)

    " $$ $

    " !#

    % ) %) ( ' % *

    ? ! #

    ! $ % ) $ %) ( ' % *

    $ % ) $ %

    + + $ ' $

    $ % + : '+ + $ ' $

    3 !

    % : ' ' $

    2 2 $ $ #

    $)( =>8-?" 7@ A7?48>" B8 7; '

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    ' % : $ '

    ' % : $ ' ' % : $ '

    7

    ) % )( $(

    )( ) $( % )( ) $( %

    " " !#

    ) & ') ) % &% % & ) %

    ? ! #

    ! $ $ ) & ') ) % $ &% % & ) %

    $ $ ) & '+ + $ * (+ + ) $& $)

    $ $ + $+ *

    + + $ * (

    6

    3 !

    #

    $+ * * (

    * $+ ( *

    " !#

    ) & )) % ' %% & ( *

    ! #

    ! $ ) & )) % ' %% & ( *

    $ ) & )+ $ % $+ ) ( $

    $ ) & )+ $ % $+ + + %

    4

    + + + % @

    8 7; '< =>8-?" 7@ A7?48>" $)*

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    $$! %3$

    . 9 $ + ++ + $$ $ +

    + $ $$ + ++ $ +

    5

    5 )

    $ 6 #

    + $ $+ $ +$ $ $

    $ $ $$ + ++ + $

    ) $ + +$ $ +$ + $

    / 9 $ %& %

    6

    %

    % #$ %& %

    %

    %& %

    %

    %

    " #

    % %& % %

    ) % +& ( +

    ) % ++ + ) % +

    2 /

    6 ) 0 % % )" 6

    / " ) % + %) %) "

    + %

    2 9 $) & % ' $ ) $% # % )0 & %0 ' * )

    8 ; '%, /#

    0

    ": 9 )$ +% $$$ % $ %) ) # ) %0 ' % &0 ) )0 0

    " 9 $%

    &

    )$'

    %)(

    # ' %0 ) & (0 0

    $), =>8-?" 7@ A7?48>" B8 7; '

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    " # ) ' &% , %%0 $ & )' * )

    " $)%

    )'

    $

    &

    )%

    :

    %$(

    !# %!

    ; '&& '&, #

    $ )% &

    ' +( *

    $ % &

    ) ( '

    ) % * $& , :

    "" # ' )0 )0 ) %)

    " # 0

    ". # 0 )0 0 )

    "/ # " 0 " 0 "0 "

    "2 # ) 0 ) 0 )

    ; '&: '') #

    $ $ )+ % &

    & + %$ ) %

    ) % + $' $ & )

    $ + + %

    ) )

    $%

    "8 # 0 0 % &

    : # 0 0 )

    # 0 )0 )

    # " 0 "0 "

    9 $ )% (

    ) % +

    0! !

    " #

    ) * ,% ( '& + $

    0 $ ) :% $ ,' ( $

    0 % & ,) * +

    9 ) '% $

    #

    ) %0 $ $ % )) '0 ) % $*

    . 9 & )$ (

    #

    ) %0 $ $ ) ) ))0 ) % (

    / 9 ( &% )

    6

    &

    8 7; '< =>8-?" 7@ A7?48>" $):

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    ! !

    2 8 #

    ) '& $

    0

    ( $% )

    0

    & '+ )

    0

    $ ++ $

    0

    ) ,' )

    8 8 #

    ) $ $+ ' )$ % &

    0

    % ) &) ' $+ ( $

    0

    ) $ &( % )& $ )

    0

    * ( '$ ) $% ) $

    .: #

    * &' %

    ) %

    & '

    & () %

    ' )

    ( %

    . #

    $ ) &

    $ $ ') * %

    $ $ $+ ) )$ % $

    $ ) %) ' $' $) $

    B$ !7 < 7 5!! $ !

    . ? #

    $ ) $ ) $) & $ ) %% ( ) ( '

    ) % ) ' $% $ ) + && ' ( ' *

    . 1 + $ ) )

    ." 1 + $ / % %

    . " A#

    ) & %) ( ' )% $$ & $)

    ) & %) ( ' $+% $+ ( $&

    .. " A#

    % ) )% : * *) ( * & *

    ) % * % () ( ' $'% $+ * )%

    $$! %3$

    ./ 9 $ )+ $

    .2 A

    $ $+ $

    $%+ =>8-?" 7@ A7?48>" B8 7; '

  • LipschutzLipson: Schaums Outline of Theory and Problems of Discrete Math, 2/e

    5. Vectors and Matrices Text The McGrawHill Companies, 2004

    .8 9 + $ +$ + $$ + +

    $ + +$ + ++ $ $

    5

    5 # 0 0 0 )0 )

    != + %

    2 %$( $+0 '$( )&0 )) )% %%

    8 % $) ,0 )$ %' $&

    ": $ $ %$'0 %$& $$%)0 )* ) '0 ( * (0 $& $) ) % $,

    % )

    " $ $)%'" 0 , )))&" 0 $')* %&0 )( %+ *

    " ) $0 % )

    " % $ )

    "" ' $+)* %&

    0

    & & %( ) &

    0 * )* $$, %( %*

    " * $&%: ),