! "
# $ $ $ $ $"
%$ & $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ' ( "
&
"
)
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ *
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ +
,
& $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ . 0
2
&
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ '/
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ '1 3
& $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ *4 & $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ *4
!
3
& ) ,
3
&
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ *4 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ *-
& $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 45
2
3
&
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 46 & $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ //
)
, 3
8
& $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ /7
! $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 6'
9 !
2!
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 64
&
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 6+
Introducción
, ,
; & ,
,
<= & & # ,
! , &
& ,
$ , , ?
, &
, ! & #
= = <= % ! . , !
,
, &
< >
, !
,
!
"
"%$ )
; . , ,
&
,
: !
& , :2 ,
,
.
!
$ 0
!
&
.
. ,
, $
@,
#
&$
A
, &
&< &
( #!& . @, !
&
& ! %$
%< & <,
&$
:':
, , @
&
&
Lenguaje Formal de la Lógica de Predicados ! !
,
& = < , & ,
= ) !
@ A &
, ,
,
, &
!
$ ! < & ,
& , ,
:# $ 2 , ! !& # ! (
'A
•
(
• •
. &
! ,
< $
<
.
;
, &
,
! !
'%
#
%<, !
'% )
!
,
&
! ∈
L
( , A !
'% ,
∈
&
; <
&
!
,
∈
L
, !
'
.
,
$ L
(
'
&
.
&
∈
A
!
&
L
.
$?
( %$ ?
,
.
$
( !
! %
( ;
$ .
( %
!
! ( , A
!
&%
!" #
& $ ? ! # ! $
&
;
.
! " #,
&
!
! , , = ( :
,
;
$
. <,
&
(
,
! ,
&
$ 2 ,
$ ,
!
$
:*:
;
( ,
$ A
•
( ;
& !
,
'%
?
,
K∈
(
<,
%?
,
,
∈
( , A
?
<
,
$
A
¬ ∧ ∨ → ↔
%?
!
A
∀ ∃
?
()
A)
# !
)A
$ ,
"
,
(
% :
$ L
:?
!
L
&
$ :
,
,
;
$ # !
4A
* "
)
!
A
!
,
*"
B
.
$
, ;
( < ,
&$
:4:
.
!
# !
+% # ! "
! ! "
:
!&
:?
¬ ¬
( &! !' !
!&
:?
!&
!
" 'A ! " / " %
%
#
! "
∀
4
!
-
!
! "* ! ="
"
6A
@, !
" ,
#
→
E! !
4C
, ∀ <∃ < !& *"
& (
&$ +A ! $
∃
!
-
!!
# !
!
$
D
!
%
A
∀
# !
"
!
!&
!&
!
"
!
?
∧ ∨ → ↔
,
!&
!
<
.
!
!
,
& $$
&
:/:
.
! ="
. "
*A
"
%∀
0
!! #
:
, !
! & <,
<
$
,
1
! !
-A ?
# !
. $
<
$
!&
!
! "
, $) $) &
!
( , < ! ,
, , !& ∃ !→∀ !! < $ ! & !& . . A ∀
0
F
<,
& <,
! !
.
G
:6:
, ! ∃
<
Otros Ordenes = ? . =.
<@
& ,
(
, &
# , , & , ,
< < @ !
!
&
!
"# & ,
%" &
%$ )
!
$ ,
4A
!
!
& !
, .
.
# !
.
<
!
&
!
A '
& % !
!
& @
% ∃∀ ∃
% &
!
+% ! A∃ ∀ ∀
" !
!
$
( , , $
& 2
.
,
/?
%$!&
, (
A
!! → "#
%$,
!=
& !
#"
!"
&
$ "
!
!! ∧
, %$
< ,
!
!
! → "#
< ,
,
$ ? ,
, ,
H ∧ → H
<
!"
.
, ,
"
? <@
<
! !&
∀
, B
" $#
,
$
$
! ,
!→
.
(
!!
, &< ,
!&
! , !
6
& ?,
&
!&
, "
,
@ ,
! %< &
%
,
& )
, , "
$
& )
)
:+:
,
) ,
$
/
Interpretación ,
#F % !
& !& $ ) . , !& !(< . , , $ ,
&<
.
,
$ !
! <
!
! !&
,
:
$ ?
, (
&
!&
<
# "
! →
6
× ×
% ,
& ∈
&
,
→{
&
,
$
&
!
$
:2
×
!
∈
:2 × ×
! <
<,
:2
×
&
$
1A
# !
!
"
< ,
#V%
B !
,
*
!
&
}
,
, & , &
,
&$
, ,
;
:-:
$
<
# "
( ( ) ∧ ( ) → ( ( ))) ∧
=∀
6A ?
"
,
!
0;
%
)
=
= '%
!=
!=
!=
%
!=
,
&
H
>'
&
#
!=
(
.
,
%
!=
$
!&
;
!=
$
!=
8(
A
A .
; ,
7
*
.
H C
.
(
,
&
A
H
! ! 7A
# !
,
!+
, :? != ' , &
K
,&I
(
!&
'
, . K
@,
(
'
K
, $ 0
!
,
"
$
.
#" #"
!=
¬(
& K
'
!
! -A
! "
!
-"
(
!
!
+
& !
,
:?
( , < ,
.
H$
!& <
& '
(! = (! =
!&
K
$ & ,
=
$? ( @,
<, ! & & ,
$ .
<.
!
:1:
,
B
$
<
:?
!
(∧)
!=
#"
(! =
)! =
:?
!
(∨)
!=
#"
(! =
)! =
:?
!
(→)
!=
#"
(! =
:?
!
(↔)
!=
#"
(! =
:?
! #"
!=
#"
!=
# ,
, !&
" ,
*∈
A
!
( * !! =
$ *∈
<( * !
!
,
%
+
, <,
!& $ 0& ? , ,
5< . ,
∃ (
!
3% &
!& !&
,
={ A A
, ,
=∀ ∃ A
& !< = = <
( (
}
=
= ! = )−
!=
& ,
, ,
A
=.
,
; $
)!
!
( * !! =
!
:?
!
∀ (
)! =
+ + ! *+,
+
:7:
! &$? ! , ,
, #,
-%$
) ∧ ( ( ))) → ¬ ( (
< & , ! & < @
! )) <
)
! =" ≤ "
A
! =" =
) @
.
= "
!& )∧
( (
,
,
,
&
( ( ))) → ¬ ( (
! ))
=.
,
, 045%
? % ?
< .
<
<, .
( ( ' '
( , < J'$ FA
,
) ∧ ( ( ))) → ¬ ( ( '
! ))
' 4
4
' 4
*
% )
.
,
J'
, <
( ( ' *
$)
) ∧ ( ( ))) → ¬ ( ( '
J*
! )) ' 4
4
* *
'
V
2
=.
%
# $
:'5:
=
A
% )
04)% ,45%
( (
) ∧ ( ( ))) → ¬ ( (
* '
*
! )) ' 4
'
*
4 *
#
=
% )
046%
<
=.
;
%
,45%
( (
) ∧ ( ( ))) → ¬ ( (
4 '
4
! )) ' 4
'
'
4 *
= !&
(
,
, &
:'':
. !=V
=
V
!&
<
"
=∀ ∃
!&
7%
={
A
!! ∧
,
!
&
A
}
=
A A .
!
/
0
0
1
1
/
) =
={
A
? , !&
V
# !
58%
# ! $
55%
"
, $)
}
. < < ! = F$ ,
!
# ! ,
;
,
,
<
. ! = V$
,
"
&
!& # !
@
!& *
@
=
<,
&
!&
.
={
!}<
!
&
,
=.
$ 5)%
56% & .
!&
@
!
!&
! $
:'*:
,
&
.
@
!
!& < . ( " 5A
# "A ? ¬ !=F,
, .
A !
!&
! , $
! V
!=V ,
. &<
@
:'4:
,
, <
& ¬
&
* . ,
#
$ %$ )
,
!& !& !
, &
$
& $) .
! = V$
<
-.
{
%
}
%
# 0
1 !=
!=
=
/. !
/
+ #, !=V 2 "/ ,
!= "
3 + V
V !/
4
{
,
{
}
}
%
.
/
5 /
&
# "%
∧
5$: ? → ∧
. !=
∧
) 1? 2
∧
&
" )A
∧ !=
∧
!=
=
∧
∧ ∧
!=
A
∧
∧
!=
. !=
→
$ ,
&
! = F$
! = F$
* !=
$
:'/:
.
, . , .
→ →
=
∧
→
@
∧
∧
,
∧
! = V(
59A ? . & ⇔ % ,
# ! ≡ ,
!&
$ ,
& <
6
#
" !=
, .
!
#
#
/ /
.
2 2
#
→
A
"
≡¬ ∨
∧( ∨
6
1
≡( →
↔
A
"
)≡
)∧ (
)
→
∨( ∧
∧ F≡ F
∨V≡ V
∧V≡
∨ F≡
∧¬ ≡ F
1/
∧
2 7
∨
/
7
/6
7
/,
8
∧( ∧ ∨( ∧
∨¬ ≡ V
≡ ≡
∨ ∨
)≡ (
)≡ (
¬( ∨
, *
∧
∧
2
)∧
∨
)∧ (
)≡ ¬
∧¬
¬¬ ≡
,
)≡
!
:'6:
∨( ∨ ∨
)
∧( ∨ ¬( ∧
≡ ≡
∧
)≡ ( )≡ (
∨ ∧
)≡ ¬
)∨ )∨ (
∨¬
∧
,
# :
"
¬∃
( )≡∀
-,
! ∧ ¬ %,&
%
()
¬
!
;% A ¬∃
!
!
2
&
.
.
AH !
A
∀ ∀
!≡∀ ∀
!
∃ ∃
∃ ∀
%
, ,
∀ ∃
!
(
!)
-, ! → %,&
!≡∃ ∃
&A3 !
!
!
, ; ! / ∃ ∀
∀ ∃
?
!) ≡ ∀ H$
"
<
%
A
! ) ≡ ∀ (¬ -, ! ∨ %,&
,
( )
¬
!!
2,
∀ ¬( -, ! ∧ ¬ %,&
( )≡ ∃
¬∀
4*
< . !$
<
"4
A
%
$
$
&
% =
∃
<
( ( ) ∨ ( )) ≡ (∃ ( ) ∨ ∃ ( ))
%0
∃
∀
( ( ) ∧ ( )) ≡ (∀ ( ) ∧ ∀ ( ))
,
,
( ( ) ∧ ( )) ≡/ (∃ ( ) ∧ ∃ ( )) <,
( , <, A '
5 0
4
"
A
,
& 5 .
4 '
5 ! 3
6 !
A∀
( ( ) ∨ ( )) ≡/ (∀ ( ) ∨ ∀ ( )) < ( 4
! $
$
, A
4 $
:'+:
.
!
=
%
#
∨∃
( )≡ ∃ (
∨
( ))
∨∀
( )≡ ∀ (
∨
( ))
∧∃
( )≡ ∃ (
∧
( ))
∧∀
( )≡ ∀ (
∧
( ))
,
%
:'-:
Formas Normales 53% ?
# ! !
.
!&
" < "
0 & < '
.
#∀ ∃%
!
!&
.
L
"
!&
/ 2
&
. !&
# ,
,
.
.
.
@
!& ) 0
, !
, @$ ?
$
, )
!&
,
!& 0
,
@$
&
# "%
& &
. )
.
. 0
% !&
" 6% ) !& .
0
<
!
$
$
! ! >
!
!
!& !&
<
@$ ,
A
:3
!
:
& # ↔ %<
,
:
,
&A
:2, & : ) !
$
$)
& <
2
!
→ (! ∧ ( →
!
→ ( ≡ ¬ ∨(
C !&
!
↔(≡
$
! & ,
!
.
$ ,
!
!&
0
( &A
:'1:
, 0
∧ /"
/ ∧ / ∧L ∧ /
!&
"=
# !
!&
57% %
!&
# !
∨ /"
¬
K
!&
5;%
K
&
&
/ ∨ / ∨L ∨ /
!&
"=
! #
!#
%
" < "
& < '
"
(
& !
# !
∧ ∨
!
"=
<
"'
'=
!&
5>%
$
"'
" < "
# ,
& <#'
"
& !
" +% ) 0 . # $
# "% (
"=
.
"'
'=
!& #
( !&
<
0
,
.
" 9%
# "% ? 0) <
,
!
,
!&
0) !
/
@
0 $
.
'
0 (
%
!&
&
!& 2?
" & <2'
?I
, &
.
$
&
I !&
!
!& !
,
#
4
!
.
& (
" < " $
!&
0 , ,
0
!&
)5%
.
!& C
" < " 0
!& 0 )
!
) ! <
!&
% .
.
# ! 0 )
$
"'
%$
)8% !& 0 ) # 0 )%< ) @$
# !
2 @
∨ ∧
!
$
:'7:
& ,
,
!
@
" & ?
"/
'$: ?
'
,
!
*$: ?
&
4$: ?
! ,
! B
"
,
!
!&
4<
∀ ∀
∃ ∀
!
. 0?
!& 4 /
$
!
@
<
&
!
!
!
$
,
<.
!$ !!
$ ∀ ∃
! JH'
∃ ∀
!
H
! J H!
!
.
!&
!
. ! ∀
, $
.
/<
! & A H7 " J
∀ ∃
!
,
∃
&
=
@
!! $
?I
! $) !
! .
<
" ,
,
∀
&
!& .
!
! %<
,
< , ) @ # 0 )%$ ) @ ; ,
( !
, < $ ? < !& $
∃
! ! %$
0
,
$
'$ .
# "% C 0 ,
?
, #
! ! & A∀ ∀
!
3% &
?
, #.
!
%$ K
!&
.
'$
∀ ∃ #% ,
B
,
$
@
,
/$: ? @ A ∀ ∀
! @ #
@
$ &
, $
!$ ?
A . .
!
7 " $
& ,
7 "
$
:*5:
.
# !
!&
))% !
(
" % ) ::
,
>% $
&
!
!
J@
<
=
!&
)
(()
∀ (¬()
5
8"
!
!
:
J@
:+
J@
,
(
A
!∨
! ) → (∃
!
!&
(
! ) ∧ ∀ (( (
!∧
)→ <
! ))) ,
A
A
!∨
! ) → (∃
(
!∧
! ) ∧ ∀ (( ( ) →
! ) ∨ (∃
(
!∧
! ) ∧ ∀ (¬( ( ) ∨
!∨
*$:
! )))
! )))
$
((¬)
∀ ∃ ∀
! ) ∨ (∃
!∧¬
4$: )
!
((¬)
!∧¬
/$: )
(
!∧
! ) ∧ ∀ (¬( ( ) ∨
,
,
!&
! ) ∨ ((
.
$?
! ))) 0)$
! ) ∧ (¬( ( ) ∨
!∧
0 <,
∀ ∃ ∀;
1
&
,
→
'$:
∀
A !
J@
'$: 3
∀
! 9 $$
:7 "J@
K
(()
%
,
;$
A
:-
: 2#@<%J @ ,
∀
0?
"
, ,
! ))) $?
0 )$
¬: ∨ ∧ ¬: ∨ " ∧ ¬: ∨¬+ ; ∨ 7 " ; ∧ ¬- ∨ ∧ ¬- ∨ " ∧ ¬- ∨¬+ ; ∨ 7 " ; 0?<
,
$
:*':
(
6$: K
!
@
$?
¬: ∨ ∧ ¬: ∨ " ∧ ¬: ∨¬+ ; ∨ 7 " ; ∧ ¬- ∨ ∧ ¬- ∨ " ∧ ¬- ∨¬+ ; ∨ 7 " ;
∀ ∀;
+$: ? , <
!
(
<
$
¬)
!∨
!! ¬)
!∨
!! ¬)
¬
!∨
!! ¬
!∨
!! ¬
# ! ,
0?
L
)6% <
!
A
! ∨ ¬( ( ) ∨
!
! ∨ ¬( ( ) ∨
!
.
←
= , $
? = '<
A L
H←
H H
B( ,
!
(
H
# ! $)
, ❏
)+% ? ! &<
!
:**:
. &
Algoritmo de Resolución Introducción 3 , ,
& ( %$ ?
'7+6$ ? #
,
! ( ,
. & ,
&
,
, $
, <
$2$
<
! , , ,
8
,
, , !
,
,
,
<
,
$ ,
,
,
&
<
,
,
!
,
&
&
$ ; ,
A
! 5
$)
< #
5
/
Resolución Proposicional -
2 " B
$ $
)5A ? B (H$ ? ,
AH
< , & 6+% ¬& ∈ < A & !=(
" "
%*-
-, $ C-
) . A H" ) A
∨
∨
&
!
.
{
}
= & ∨& ∨& ∨ ∨&
# "% ?
@
.
&∈
,
− {¬&}) $ ?
$
B (H$
∨
.
" 3&
C-
A ( → ∨ ≡ ¬( ∨ ∨
# !
− {&}) ∪ (
-
, H H? ,
A (∨ ∨
! !
"
$
*
(
'%
= ¬& ∨ &
) ∨&
∨
∨&
$ ,
&
:*4:
! =& ∨& ∨ &
∨& ∨& ∨
∨&
)
∧
(
→ &
∨
*< ,
{
.
)
}
$ ? ,&
(
.
)
@
''
∨M ∨
∧¬ ∨
'
*'
. <
< !&
∨M ∨
→
*
''
<
∨
'
*'
∨
*
,
, <
$
# "% ? ; <, < !& , &
< ∧
A
*
→❒
< ∧
$)
, (
& <, !
❒
.
.
$
& =.
<
(
!
. $
<,
$? !
$
<
>
,
(
$
" (
% 2
7 !
7
%
∈
'$: 8 *$: ?
6$: ?
"
!
&
/$: ? 4$: ?
.
@
.
, .
&∈
¬& ∈
A 4$:
,
∨M ∨
* !
@
& .
.
E
& < !& $
*
" 7% ?
<
, =
A
D
)
. ,
>
!=❒
&
<2>
&
(
.
2@ C !
7 B
A 2@ C ))% ? 7 . 7
7 !
'
. !
7
7
!
{
( ,
&$)
:*/:
¬ ∨ A
¬ ¬ ∨¬ ∨
}<
#¬ %
:? :?
=
:?
=
)
# ¬ ∨ ¬ ∨ %< #¬ ∨ % !
,
! 74O$ 2
<@ N?,
<
$
, ,
& , , $? ,
&
, 0 !
& , N8 :2 ;
, !
! =
)6% ?
"
❏
<7 &
$ A¬
,
.
2
¬ ∨¬
,
&
7'O ( 7 (
! !
= { ¬ ∨ ¬ ¬ ∨ ¬ ∨ } <, ( . !& < ∧(¬ ∨ ) ∧(¬ ) ∧ (¬ ∨ ¬ ∨ ) $ , A
. < ! $
¬ F
¬
F ¬ ∨ !
¬
F ¬ ∨¬ ∨ !
F ¬ !
" 5% ?
( , !
. #, $
,
# "% )
<*
!
$ (
,
.
( .
$
# "% )
" +% !
!
! .
=
<
! !
7
" 6% ,
# ! =(
! <
! <
" )% *
!
7 ! ! , & , . 7
69% ?
< , &%
$2 <
<
! ! $ ! <
.
&
<
@
$ .
!
!
<
;
=( $
:*6:
!
" 9%
,
=( $ <,
& ! !
=
,
<
$ !
# "%
.
<
.
.
A
" ¬ ! =F
! =V ' F
:)
. %<
! !&
'
<
9<
,
= ¬ ∨ ,# 9 ∨ ,# 9 0 =
,
,
.
'
.
%$ ,
'
0
A
! = ,# 9
'
∨ ,# 9
0
0
<,
'
,# 9
<
'
$
%
'
< ,# 9 '
$) "
0
<
<
'
0
! = ,# 9
!
,
0
<,
{
#
%#
%$
"
,
,
9
0
0
'
!
. ¬
&
&!
%
)
,# 9
;
'
'
:) , .
0 F
%!
¬ ∨
¬ ¬ ∨
.
}(
!
! ! <
, ( A
( , < <( ¬
¬ !
F
¬ F ¬ ∨ !
¬
F ¬ ∨ !
F ¬ !
! !
<
+ *$
!
-
:*+:
#¬ % .
! !
/,
'
∨ ,# 9
0
" ;&
"
@
" <,
!
'% ? &<
C
(
$ # "% ! # , !
(
!
'%<
<
@ =( # (
, ,. > . ( , $
#, <@
!
! 6%
#
/%
>
(
$ ?
<
.
!
% < <$$$< ,
! .
,
Resolución General =
<
&
,
, & & , ,
& , , < ! . $
!
&
, . $
& ,
!
!
, ,
<,
& .
! &
,
$
2 & ,
. <
, (
!
@,
, @, <=
!
!&
$
&< .
@,
$
# ! {- : - :
63A
σ
}
- :
( <
-"
- - L -
-"
$
? σ(
σ(
)=
$
<
0
(
&
&
"
) )+A !
"
"
6;A ? σ = {- : - : - : ', σ < , σ ( )< , & -"
# !
"
67A 0
!
$ *
# !
!
}
σ( ? !! $
=
!!
:*-:
@, & < @, & . ',
"
)
! #" = K
'
*
σ ={ :
:
%$
!}
σ ={ :
6>% ? <
# !
:
"
{ :σ ( ) :σ ( ) } ' ∉{ σ ={ :
!"- .
σ ={ :
σ
!
:
:
}7
, , " & σ$
:<
: #'}
,
&
?
'= K <
( ") .
:σ
:
,
&
?
=
σ
!=
!
"
}/
!! <
σ σ
} σ
:
σ (σ
!!
!
A σ oσ
σ
σ
@, .
.
)7A
σ oε = ε oσ = σ ,
={ :
:
@,
$
}$
A
!
+5A
.
,
&
$
""
)3A ? σ = { : σ ={ : ! : :
( " )$
σ σ σ
,
%
=σ
(
,
<
"
}
( )) = σ σ ( ) ,
-
7"
, , ε$
,
"
# !
!
:# } &A
:#
σ oσ
:
&
,
: σ (σ
σ
={ :
σ ={ :#
% σ o σ oσ ! = σ oσ ! oσ ,
:@
1
} σ
&
) ,
'
"
!
+8A
# ! .&
( )} {
:σ
}
:
!) =
!=
=σ
! !
<
( )
!!
!!
<
=σ
.
( )$
.
@ '
' !
! !
,
!
: $
:*1:
%! ! $ ?
<
!$
!
# !
{ ,
:
+)A ? : ,
}
" .
@, & "
{
( '
, , }⊆
'
:*7:
.
, &
' ! .
Unificación , 3 @, ) , <
# !
! & ,
< '7+6<. , &$ <
.
NP
$? , .
< =
( .
!
@, ! < A ω(
ω
$
={ < .
);A ? !
?
<
! !
( , ! ω
{% : -}
. A ω =ω
, (
,
ω(
) = ω(
{
( <
!
, ) = ω(
} ,
)=
= ω(
@,
>A ? ω ω
!
:
!
1!!}
}
!
, .
$
(
{- : %}<
,
!
@,
.
! < ,
:
%:<
<
$
:45:
,
ω′
@
!
, !
- !!}<
( !
:
! !
! !
ω ω
,
)
ω ={ :
A ω ={ : $?
< , ; , ω =ω
={
$
,
, .
.
. : }, , . ω ′ = ω oσ $
! -:% : } , {- : %} .
$ "
'745< < , , ! $
,
,
( , )>A ? ={ ! % !! ω ={ : ! %:- : } ! ω′ = { : ! $) ( , ! ω , & σ = { : }$
,
! &
!}< )=
ω, ++A ? . ! ! "* # % , & σ . , ω ′ = ω oσ $ ! ! ,
!
&<
= 17O$
+6A & ω$ ? &< (
# !
,
! > , 3 !
"
.
L
@,
7<
. ω {% : -}< ,
2 (
&
, @, .
! !
" @,
@, @,
!
< ,
$ ! $ ?
(
=
< ! < , &< < !
, ,
!
$ ? ,& , . = = ! ;
, $ ,
@, $
!
,
(.
!
.
.
!
. @, @,
@, @,
$
! ,
<
$ &
$ ) , # !
, &
!
.
(
$ (
+9A
@,
={
,
,
# (
! :?
!=∅
A :?
@,
?
! '!
={
'
!=
68A ?
={
'
'
}
!
A
!}
! !=
'!
,
9
.
$
!
! ! <
!
<
≠∅
(
,
! '
?
'5?
H'5<
H
={
" @,
!
A
<
:?
}< ( )<
@, &%
$?
! ,
!
! !} <,
! (
'
. ,
A
<
. ;
:4':
!
,
# <
≥'
={
!} <.
={
!
<
}<
@,
!=
<. 9. @
, .
"
'!
, , ={
≠∅
!
(
$
<. <
}
* <,
<
! ={
!=
<
!
}
!
& A 0-= ' <σ ' -= ε
'$: *$: ? σ 0
!&
!
&
4$: ? A
0
0
-=
!
, σ 0 Q%
σ0 .
A? A
#QL
(σ 0 ( ))
@
$
{- : }
σ 0+ = σ 0 0 =0+
B
,
A ?
.
! . , !
, & ;
@ . !
*$ (
& , , ,
!
$
& $?
= , $ ,
@, &
:4*:
. ,
4
, .
< . $
4< < &
"
65% 2
={
!
!}
!!
5% 0 = ' ( σ ' = ε )%
'
={
!
} , =
-=
6% 2
!={
)% σ
!
σ ={ :
!
!!
!}( 0 =
!!}(
!
={ .
6% <
"
={
6)% 2
!!!
!}
!
0"
!
$
!!}
!
5% 0 = ' ( σ ' = ε )%
'
={
}
6% 2
4/( 4
σ ={ :
!={
)% σ
}( 0 =
!!! % 40( 4B&,' σ = { :
6% 2
! ={
)% σ
!
6% 2
!!!
4,( 4 & ' ! ={
)% σ
: !
σ ={ :
={
!!} (
!
:
!} ?4) !}
={
!!
!!!}
!!
!}
!}
:
2
!} ( ?46 σ $
,
*
" ,
4
! $
2
&
=. ( ,
&<
=. H =.
$
:44:
. H#
= I%$
"
={
66% 2 )
'A 0 = ' <σ ' = ε $
)
*A
)
4A ?
)
*A σ
! ={
)
4A )
.
'
={
! . <
} 4 "4 σ = { :
3 ,
!!}$
!
,
(
6+A ?
={
!!} <
! ,
& & , (
}<0 =
<
={ (
!} 0"
!
.
$
" ( !
@, $
! & , & ,
!
! & @,
$
,
$ "
!
. !
!
'
'
:4/:
!
!
−
−
!}$
Algoritmo de Resolución General & ! & !
)
" ,
, # !
& $ 2 !
. &
,
.
.
, @,
&
.
$
+3A
¬& ∈
, &
,
!
ω$ ?
,
!
*
<
.
& ∈
.
,
& , &
A ! = {ω
& &
"
! − ω & !} {ω
69A =
!! ∨
!
!∨¬ & =
ω ={ :
!
:
=¬
! !
63A ?
"
? < !¬
!}
, !
ω ={ :
!∨¬
A
}
&
!! ∨ =
=¬
A
& !!
!
. < ( !
!
!$
. ,
#
$
< $
∀
!&
! , !
{
!! ¬
!∧¬
. !$
(
, $ /*%< , ,
(
;
( , <
! 67% ?
!
( ! ,
&
"
!!
! =❏
) . ;$ < , .
!!
! !
}
!=
& &
!! ∨
!!
& =
!!
<
{
! − ω ¬& !}
!} $
:46:
< = < < $? ! ( $
=
=¬
!!
,
!
=.
.
$?
!
<
<
A -
=
-
!!
=¬
!
ω ={ : ) !
!}<.
. < < <
∀
. < A
!! ∧ ∀ ¬
( 4
<
A { (
(
-
A
!!
-
! =❏
{
(
!}
<
!
< <
<
''$
!
5
#
!} $ ! &
!! !
% # ) ! #
;
#
# 3/ 6
-
@
" (
% 2
7 !
7
%
∈
'$: 8 *$: ?
A 4$:
>
! !=❒
/$: ? 6$: ? 4$: ?
(
, $ ,
<2>
!
, !&
.
2@ C 7 B
A 2@ C
''
7 !
' .
! ∀
7
7
!
< & !! ∨ ¬
:4+:
!
=. A
!! .
$
!! ∨ ¬
!$
6;A 2
" ¬
2
&
!∨
!∨
!! ¬
!¬
!
,
!∨2
!∨
! ¬2
(
!∨
!∨¬
!!
! ¬2
!∨¬
':#
1
#
; ¬
!∨
¬
*
!∨
!∨ 2
!!
!
/
!
6
¬
!∨2
+
¬2
!∨¬
!
-
¬2
!∨¬
!
1
¬
7
!∨
!
! !!
{
:
}
−<
{
:
}
−)
ε
!!
'5 !∨
''
{
!!
'4
2
'/
¬
'6
:
=−(
}
−)
ε
!!
'*
!! !!
=−
{
:
!}
>−
{
:
!}
?−
ε
❏
'− )
. (
, ,
!
4
.
!
#
,
'*0
&
!!
!∨
4
,
$2
%(
$ '
)
!
(
'6% ,
A
. &
.
,
.
,
!
, $
:4-:
'*$
$
<
,
,
'% ?
" 58& & # "% )
. .
& & '
@, , .
& σ ,
&
ω( ω( 2=
σ(
&
)
,
)$ ω(
.
&
ω
! = ω & !∨ω
ω
! = ¬ω & ! ∨ ω
=
? ,&
& &
.
) ∧ ω( ) = ¬& ∨
.
?ω
,
#3
.
& ∈
ω & !=ω & !
&
! ! J ¬ω & ! ∨ ω
!
< ! = {ω
! − ω & !} {ω
! − ω ¬& !}J ω
&
!∨
<
A
F
/
¬& ∈
.
$
.
=>
&$ .
N'O
,
)
@, <,
'"
& ω
&
=& ∨
A
! ,
)
,
?
$
∨ > )7/ ?∧ =¬>
0
∨ > )70 ?→ > )7/ ∨ > )70
F
:41:
!
<
%
ω &!
F<
,
V %$
! = $)
.
@,
&
, V
< , , < !&
= = !
,
!
# @,
. &
F
,
$?
, . A
ω(
)∧ ω( )
<
ND
ω( ω(
E:
A
) )
.
&
. <
O
!
, !
(
, <
$ ,
&
:
L @
:) V : F$
(
<
(
!
!
. ? =
, , , ,
A !
< .
$
,
, ,
,
&
!
(
#
,
%$ " 55 &
σ
.
′=σ
ω′
′ =σ(
ω
σ
'% ?
"
)
ω
′ =σ
!
σ
; %
! ′
@
#,
.
# ′
%$ B
.
!
A
ω σ
′=σ
σ
′ =σ
!
σ
σ
!
′=σ
ω′
!
′ =σ
σ
′
′
:47:
!
:
( (
<,
A ? ,&
. A
= {& ∨ & ∨ = {¬&
}
∨&
∨ ¬&
+
@ , &$
,
∨
+
∨ ¬&
.
&"
<,
+
}
!
.
.
<
, >'
@
> '$
?
,
. A
′ =σ
! = {σ & ! ∨
′ =σ
! = {σ ¬&
∨ σ & !} σ +
!
∨ σ ¬&
!∨
ω′
{σ
A
+
!} σ
!
σ & !σ &
& !
+
σ &
!
+
!}
A ′ = ω′ σ
)
{σ {σ
!! U ω ′ σ
. σ σ & ω ′ (σ ∪ σ
(σ
∪σ
){&
∪σ
& !
& !
σ
R
& &
(
σ( σ ω σ oω
! &
+
!
+
!
=ω
<@
!
)%
& @
)=σ
+
σ
{&
; σ & & +
!
!
σ ∪σ
&
σ
&
+
}=
& ! σ ∪σ
!
ω ′ (σ ∪ σ ?
)
σ ∪σ & !σ &
? @ <, ω ′ (σ ∪ σ
N*O
!!
&
& +
+
!}
+
}
,
!
. A
!} =
, ω′ $
{&
< ω , . <, & σ .
& & !
.
&
+
N4O ,
!
ω<
A
)= !∪ω ∪
+
#
ω
!∪ω
<
!! = !=
# ,
:/5:
N*O%
}<
ω ′ (σ ∪ σ
)(
∪
ω ′ (σ ∪ σ
)(
) ∪ ω ′ (σ
#)
)J
. !
σ ∪σ
#
!=σ
J ω ′(σ # ,
)(
∪σ
!
(
σ ∪σ
ND
;< <σ
! !%
!=σ
)) ∪ ω ′(σ (
)= !
& σ & ! %< ,
A
)) =
E:
O%
!
J ′ ) "
σ(
)=
= 6>A ? A σ ={ :
′<
< ′ !∨
!
! 3! ∨
:
!}
!
σ ={ ′:
:/':
=¬
′:
}$ ?
′!
′ !! ∨ ¬
′
.
′!
A
=
σ =
l
:
!
:
!∨
!
:
!
′=
! !∨
q
σ =
!! ∨
!
′!
=¬
!
m′ :
!
!! ∨ ¬
′! ∨ ¬
′
ε ′=
? ω ={ :
!
: =
′!
:
!∨
&<@ ′! ′ :
}.
! !∨
!
!! ∨ ¬
!
.
A
′!
=¬
ω
σ
σ
=
′=σ
′ !! ∨ ¬
σ=
!
′=
′!
m′ : r
!! ∨ ¬
′ =σ
" 5) & " ! < $
, <,
!
!
& .
, ,
&
:/*:
%A ? &
′
′!
r
′:
′ =¬
!!
′! ∨ ¬
A (
′!
!
A ? 7 .
(
!
!
7
#, @
L
(
!
#
,
@
& , ,
&
@ < ? !¬
!!}
% $
7
&% (
; 7
.
!
(
& @A
% 7
#
{
<
A
,
$
7
& <,
,
,
&<
,
;$ < !
, <
<
.
:/4:
!
< <, !!
( , < !
(
A
Estrategias de resolución 1
0 $
5
#
3
* ; ;
% 5 #
/ #
3
/
, - @A 0 B/
#
0
5
8 % #
5
3!
/
4
%
5
$
# 3/ .
#
$ ; $
/ 6 %
# $ /
Estrategias de Borrado 4
#
# $
5 # 1
/.
0 ;
#
3
#
;
/
#
# ! !.
+7% ( .
@
%
;
, ;.
<
,
%
&$) , $
& "
={
( <,
+8% <
!
.
$
. ,
&
.
, &
*
# !
!¬ !∨ < ,
!! ¬
!∨
!¬
+;%
.
.
∨¬ ∨ ∨¬
+5%
$
$8$
,
( !
(
,
.
$
!} <,
$ !
"
<
!
(
,
>
$
!
!
://:
! .
<
& ! (
$
,
<, ,
. " ,
<
( ;.
!
. &
,
$ 0
, .
"
!
1
!∨ (
!
& { : < ,
, $ (
{
(
!
!}
!¬
!
! $
!
# ! 3 @
.
+6% !∨¬
,
,
,
$
{¬
(
+)% ,
"
(
@
,
, $
7
+>%
$? "
& σ
.
σ
!∨¬ ! ! ∨ ¬ -! ∨ 1! ¬ ! ∨ ¬ -! ∨ 1! , . : -} , < <
<
!⊆
$
!
!} $
!
$
? , <
. (
( 3 , ( $
3 $
!
&
! . <
! <,
<
, .
=.
, (
&
&<
,
$8 ( &$
C # ! ,
98% #
&
%$ ,
&
&
$ "
'$: *$: 4$: /$: 6$: +$: "
={ ∨ <, $
++% ? &
¬ ∨ <
∨ ¬ ∨ ¬ ∨
-$: 1$: % 7$: '5$:
¬ ¬ ¬
)! )!
&
.
( $) , , .
∨
,
>!
. &
= ,
¬ }$ 2 ,
¬ ∨
( , <
. , ' *,
$? .
:/6:
$
,
&
,
<
<
!
& . .
$"
< ! !
,
;.
$
=
$
! < ={ ∨
<
, < , ∨¬ ¬ ∨¬ } , .
¬ ∨
)
< , ! $
@
%
! $
&
) !
( , < <
<
( & $
! ,
L .
#
(
<
L
,
&
Resolución de Entrada # ! ,
95% (
={ ∨ ¬ ∨ <, < (
+9% ? &
" .
$ , $ ¬ ∨ ¬ }$ 2
&
& &
,
, A
, '$: ∨ *$: ¬ ∨ 4$: ¬ ∨ /$: ¬ 6$: ∨
-$: ¬ 1$: 7$:
)!
!
+$: ∨ ? , , <
!
.
& .
!
& ,
$ .
L
<, $
,
& ( , < ,
$
Resolución Lineal 8
0 !
5 #
% /
:/+:
,
, ,
9)% ?
# !
(
(
7
2 ! #'%
=
'
"+
$
* .
,
4@" /" "A/A
"
#
"
%< #*%
,
"
#
%
,
"$
&
,
$
& , ( < , (
/%
* !
,
.
.
,
! #
(
={ ∨ < ,
, ∨
¬ ∨
. ∨¬ ¬ ∨¬
¬ ∨
}$ "
$
∨¬
¬ ∨¬
¬
❒
&
=
;
,
, (
? ,
.
&
,
$ $2
<
( ∪
! &
.
lq
!
< $)
(
!
, (
<
<
.
,
$
Resolución Ordenada *,
8
*
#,&, "-
#
%
5
/8 #
/ 8
#
#
3
! #
3
!/
:/-:
={ ∨ #
+3% ? & %A
" !
¬ ∨ =
¬ }$ 2
¬ ∨
&
, ,
'$: ∨
6$: ∨
!
*$: ¬ ∨ 4$: ¬ ∨
+$:
>!
/$: ¬
-$:
)
6
; .
, &
, 6< /,
' , * 4, $
,
& , ? L
& %$ < =
#
! .
( , < < & &
'
4
,
$
$)
,
,
$ , ,
; ) #2
/ ,
!
/$
, 4 /$ +<
,
,
,
&< &
( &
> .
& &
#
7
L & %.
$
,
< & ?
!
:/1:
&$
Prueba de Teoremas por Resolución , '4
"
! =.
,
& !
. , @ .
$
,
!
, (
( !&
, ,
,
@
<
.
'74'< S
,
&.
<, &
& , < $
<
! , ,
,
!& ,
$? , , =
<
,
,
, ;
, 5< ! *%
+ "
&
#
! ="
! =" + "
⋅ "
, ,
$ A
' !
*$: ¬#%
! ∨ #%
*%
*%
, & $
&
& #
(
# ! # !)
+
+ !=
+ !+
' '!
/$: ¬
-! ∨ ¬#%
- !∨
*%
& .
( , # & &
=
<@
&
! ="
$
$
@
( = ,
'$: #%
!
$
.
,
( ,
+7% 2 &
,
@ &
!
.
'4
@,
; !&
4$:
$ ,
,
"
&. !&
(!
$ '755
@ ) #%
$ ? @
( &
%.
!
&
,
@
#
!
!&
0
& %
# @
&
?
"
$
, !&
% %
:/7:
,
(
#(
, . !
) )
∗
+ != ∗ +
<
,
& , ,
2,
@ ;
,
$ )
,
< , ( , <, $ ? ,
@
; &A
T ,
A
∃ <
(
) ,
*%
(
# (')))
,
<
>
&
&
<
:65:
& . @
( ;
$
Bibliografía
CD E6
N
( F */ D E6 . , " &+ " + 7 G 2 /
+
% , + ", , (( !
16O $ 2
CH
#'716%
; =?F */ I/ H " &+ % * " #+ + * K 7
N?" 74O *) 3! NP
!
= 17OP 2 C
?" + * /"+ A ?"E P
= 6 B ,
N
C*
N0
7*O? 3 2 1-O
'774
%
'717$ ,,$ 7*:'*/
$ &
P 3 :U
= U ?, B
(=?!
7 3 2 ' ? ** U '-$ 0''5$ C
?
NP V I -7O3$ P V I $ * 0 =L $ 2 ! $ #'7-7% NP
; /J/ " " " &+ ,&&" , , ; 2 /
,> A &
$* $
<)
&
$ #'77*%
$* $ $* $ $
==F,
* , % " +1" + M; D % :
75O !0 < C $ "* $ $ = U W?
$ #'71-%
./ L " ++ 7
*
;
2 /
(==!
I $ #'775%
:6':
C.
; 8 6
N?
( FN/ . -6 EL
;
H/ 6 .
7
;
(( !
1+O ? < = ?= , !5 * $ = C ) $ * $ $ 7/
#'71+%
:6*:
Índice Alfabético H =.
H
44
,
*
! @
&
*1
"
/7
"
4
" 4+ (
@
4'
" (
*/ "
4' !
4'
!
2
,
!
&
*
4* 4 /
*"
." *
*
4
* "
." *
/
**
*
& =.
'/
4+ &
'*
'1
0
'/ *
/-
/
4
! *
**
A
**
* #
3
4+
*
.
**
&
/7 -
" 46 .
&
'6
*4
*
& ,
46
,
47
"
@
*&
,
&
@,
&
*,
*-
"
C ,
@,
//
, /*
:64:
@,
4'
0
'7
<
'7
0
'7
<#
'7
0)
'1
//
0?
'7
*
"
*'
*
'7
*
@ '7
+
0
(
0
(
0
)
?I
'7 ,
'7
'7
" +
0
'7
0
)
@ '7
+
2
* 0
)
@
'1 "
! "
'1
/
/
"
'* 6
! "
*6
! !&
!
'* **
!&
!
'* = I
!&
44
'* '1
! !&
!
/ ,
S
& & /
A
'
/7 '/
"
**
B
/ L
45
L
/7
& 3
& , ,
*4 *4
**4<46 *
*-
&
-<1<'/
/+ ,
&
3
45
2
4
/5 '7
:6/:
&
*-
&
*-
&
*7<
&
*1
4
"
45 !
&
45
45<46 !
45
!
45
/6 !&
1
//
.
!
/
, , , L " .
!
47
/
47
/
4/
45
/7
*1
:66:
Información de Contacto
* ! "* G #
. '
&
"
VVV$
= , *
!
/
:6+:
$
$
X $
$
X
$