IDEPUNP / CICLO REGULAR / SETIEMBRE-DICIEMBRE SETIEMBRE-DIC IEMBRE 2011
1
SEMANA Nº 09 TEMA: Proposiciones Categóricas
Coordinador: Lic. J. Martín R!" R!"
1.
#$r%a& #$r%a&i'a i'arr n ($r%a c)anti c)anti(ic (icaci aci$na $na&*&* +Nin, +Nin,n n ani%a& ani%a& c$n c$n c)r c)rn$ n$"" " carn carníí$r$ $r$. . Sa Sa A ani% ani%a& a& c$n c$n c)rn$" B ani%a& carní$r$. a ∀3 ( A 3 ¬ : B 3 ) 4 ∃ 3 ( A 3 ∧ : B 3 ) c ∀3 ( A 3 → : B 3 ) 5 ∃ 3 ( A 3 ∧ B 3 )
A 3
∀3 ( :
→ B3
)
2. #$r%a $r%a&i&i'a 'arr n ($r ($r%a c)a c)anti nti(ica (icaci ci$n $na& a&** +A&, +A&,)n )n$" $" 6r$( 6r$(" "$r $r" " )ni )nir r"i "ita tari ri$" $" n$ "$n "$n c$n( c$n(r rn nci ci"t "ta" a" 5irti5$" a ∃ 3 ( U3 ∧ D3 ) 4 ∃ 3 ( U3 ∨ : D3 ) c ∀ 3 ( U3 ∧ : D 3 ) 5 ∀ 3 ( U3 → : D3 )
∃3(
U3
∧ :
D3 )
7. #$r% #$r%a& a&i' i'ar ar n ($r% ($r%a a c)an c)antiti(i(ica caci ci$n $na& a&** +T$ +T$5$" 5$" n$ "$n "$n a&)%n$" a ∀ 3 ( : A 3 ) 4 : ∀ 3 ( A 3 ) c ∃ 3 ( : A 3 ) 5 ∃ 3 ( A 3 )
: ∃3
( A 3 )
8. #$r% #$r%a& a&i' i'ar ar tí6i tí6ica ca% %nt nt* * +T$ +T$5$" 5$" &$" &$" 9r4i 9r4itr tr$" $" "$n "$n n$ :),a5$r" ;;;
a A J 4 A J c J A
;;;
5 A a J ;;;
A i J
<. E& =)i =)ia a& &nt nt 5* 5* +A& +A&,)n$ ,)n$"" %t %ta& a&" n$ "$n "$n n$ c$n5)ct$r" n ($r%a tí6ica "* a C aM 4 C $M c M$ C 5 Ma C M$ C
. Si +A&,) +A&,)n$ n$"" &,i &,i"& "&a5 a5$r $r" " "$n ci)5 ci)5a5 a5an an$" $" " (a&"a. (a&"a. C)9& 5 &a" "i,)int" 6r$6$"ici$n" "$n (a&"a" 1. A&,)n$ A&,)n$"" &,i"& &,i"&a5$ a5$r" r" n$ n$ "$n "$n ci)5a ci)5a5an 5an$" $" 2. Nin, Nin,n n &,i &,i"& "&a5 a5$r $r " ci)5 ci)5a5 a5an an$ $ 7. T$5$" &$" &$" &,i"& &,i"&a5$ a5$r" r" "$n "$n ci)5a ci)5a5an 5an$" $" S$n cirta"* a 1 ! 2 4 2 ! 7 c 1 ! 7 5 S&$ 2 S&$ 7 . Si +Nin +Nin, ,n n c$% c$%6)" 6)"t$ t$ $r,9 $r,9n nic$ ic$ " )n %t %ta& a& " r5a5ra. C)9& 5 &a" "i,)int" 6r$6$"ici$n" "$n (a&"a" 1. T$5$ c$%6 c$%6)" )"t$ t$ $r,9 $r,9nic nic$ $ " )n )n %ta& %ta& 2. P$c$" P$c$" c$%6) c$%6)"t "t$" $" $r,9n $r,9nic$ ic$"" n$ "$n "$n %ta& %ta&" " 7. Ca"i Ca"i t$5$" t$5$" &$" c$%6) c$%6)"t "t$" $" $r,9ni $r,9nic$" c$" "$n %ta %ta&" &" S$n cirta"* a 1 ! 2 4 2 ! 7 c 1 ! 7 5 S&$ 1 S&$ 2 10. Si +A&,)n$" "acr5$t" n$ "$n a4"t%i$" a4"t%i$" " (a&"a. In5icar & a&$r 5* 1. P$c$" P$c$" "acr5$ "acr5$t" t" "$n a4"t a4"t%i$ %i$"" 2. Nin,n Nin,n "acr "acr5$t 5$t " a4"t% a4"t%i$ i$ 7. T$5$" T$5$" &$" "acr5$t" "$n a4"t%i$" a ##F 4 #F# c FFF 5 F#F FF# 11. 11. Da5a &a 6r$6$"ici 6r$6$"icin* n* +L$" 6$"t)&an 6$"t)&ant" t" a &a )nir"i5a5 )nir"i5a5 n$ "$n a&)%n$" rcin ,r"a5$" 5 "c)n5aria "c)n5aria " =)ia&nt n ($r%a tí6ica a* Sin5$ A* 6$"t)&ant" a &a )nir"i5a5 ! B* a&)%n$" rcin ,r"a5$" 5 "c)n5aria. a B a A 4 A $ B c A i B 5 B $ A A i B 12. La n,acin 5 &a (r%)&a =)ia&nt a*
E& =)i =)ia& a&nt nt 5* 5* +P$c$" +P$c$" n$ ci)5a ci)5a5an 5an$" $" "$n n$ &,i"&a5$r" "* a Ba"tan Ba"tant" t" n$ n$ &,i"& &,i"&a5$ a5$r" r" n$ n$ "$n ci)5 ci)5a5a a5an$" n$" 4 A&,)n$ A&,)n$"" ci)5a ci)5a5an 5an$" $" "$n "$n &,i"& &,i"&a5$ a5$r" r" c T$5$" &$" &$" &,i" &,i"&a5 &a5$r $r"" "$n ci)5a5 ci)5a5an$ an$"" 5 Nin,n Nin,n n$ ci)5a5 ci)5a5an$ an$ " &,i"& &,i"&a5$ a5$r r Jai Jair r " ci)5 ci)5a5 a5an an$ $
1.
∀3
( R3
2.
∃3
( R3 ∨
7.
∀3( :
?. La c$n c$nr" r"a a acci acci5 5nt nta& a& 5* 5* +Nin +Nin, ,n n at& at&ta ta " n$ 54i&)c@$ "* a A&,)n$ A&,)n$"" n$ n$ 54i 54i&)c &)c@$" @$" "$n at&ta at&ta"" 4 T$5$" &$" &$" at&t at&ta" a" n$ n$ "$n "$n 54i& 54i&)c@ )c@$". $". c La %in$ %in$ría ría 5 n$ n$ 54i& 54i&)c@ )c@$" $" n$ "$n at&ta at&ta"" 5 P$c$ P$c$"" 54i 54i&) &)c@ c@$" $" "$n "$n at& at&t ta" a" A&, A&,n n n$ at& at&t ta a " n$ n$ 54i 54i&) &)c@ c@$ $
8.
∀3
>.
→ S3
:
R3
( S3
)
S3 ) S3 )
→:
→ R3
S$n cirta" 3c6t$* a 1 ! 8 4 2 ! 8 c 7 ! 8 5 S&$ 2
)
∃3( :
R3
∧
S3 )
"
IDEPUNP / SUPERINTENSIFO 200>
RAHONAMIENTO LGICO
1 ! 2
17. La 6r$6$"icin* +N$ @a! n$ 6$&ític$" " =)ia&nt a* 1. A&,)n$" "$n 6$&ític$" 2. T$5$" "$n 6$&ític$". 7. T$5$" n$ "$n 6$&ític$". 8. E" a4")r5$ =) n$ t$5$" "an 6$&ític$". <. E" (a&"$ =) n$ t$5$" n$ "an 6$&ític$". S$n cirta"* 7 8 ! <
c
∀ 3! ∈ ¡
( −3 − 7 ≥ ! ) ∨ ( −2! + 3 < 3 )
5
∀ 3! ∈ ¡
( 3 + 7 ≥ ! ) ∧ ( 2! − 3 < 3 )
∀ 3! ∈ ¡
( 3 + 7 ≥ ! ) ∨ ( 2! − 3 < 3 )
1. La $4r"a 5* +E" %ntira =) &a %a!$ría 5 :n" n$ "$n n$ $ci$"$" "* a P$c$" :n" "$n $ci$"$". 4 Nin,n :$n " $ci$"$. c Ca"i t$5$" &$" :n" n$ "$n $ci$"$". 5 C)a&=)ir :$n " $ci$"$. Nin,n :$n n$ " $ci$"$.
S&$ 1 ! 2 c S&$ 7 1 2 ! 8 S&$ 2 ! 8 18. La c$ntraria 5 &a c$ntra5ict$ria 5& ")4a&trna 5 +Nin,n A " B "* a T$5$ A " B 4 Nin,n A " B c A&,n A " B 5 A&,n A n$ " B T$5$ B " A 1<. En &a &,ica 5 6r5ica5$" &a 6r$6$"icin +Si J)an "t)5ia nt$nc" tri)n(a " "i%4$&i'a c$%$* a 6 → = 4 J
21. La $4r"a 5 &a c$nr"a 5 &a ")4a&trnant 5 &a c$ntra5ict$ria 5 &a c$ntraria 5* +L$" 5%crata" "$n 6aci(i"ta" "* a A&,)n$" 6aci(i"ta" n$ "$n n$ 5%crata". 4 K)in=)ira =) "a 5%crata " 6aci(i"ta. c M)c@$" 5%crata" n$ "$n 6aci(i"ta". 5 Nin,n 5%crata " 6aci(i"ta L$" 5%crata" n$ "$n 6aci(i"ta". 22. La c$ntra6)rta 6arcia& 5* +La" %5)"a" "$n @trtr$($" " "i%4$&i'a n ($r%a c)anti(icaci$na&* a ∀3 ( : 3 → : M3 )
→ Mt
c D : ∧ E% 5 E :
20. La c$ntra6)rta t$ta& 5 &a c$ntra5ict$ria 5* +La %a!$ría 5 in,nir$" "$n 5$cnt" "* a A&,)n$" n$ in,nir$" "$n 5$cnt". 4 Nin,n in,nir$ " 5$cnt. c P$c$" n$ 5$cnt" n$ "$n n$ in,nir$". 5 Cirt$" in,nir$" n$ "$n 5$cnt". Nin,n n$ 5$cnt " in,nir$.
→T:
c
( 3 ∧ M3 ) ∀3 ( M3 → : 3 )
5
∀3 ( :
∃3
4
E : ∧ T: 1>. #$r%a&i'ar* +T$5$ n$ :),a5$r n$ " at&ta a : ∀3 ( : J3 → A 3 ) 4
: ∀3 ( J 3 → : :
A3 )
c
∀3 ( :
J3
A3 )
5
∀3 ( : :
: ∀3 ( :
→: :
J3
→:
A3 )
1?. N,ar &a "i,)int 6r$6$"icin cat,rica* + ∀ε ∃δ / ( 3 < δ ) → ( ( ( 3 ) < ε )
) ( ( ( 3)
>ε
)
) ( ( ( 3)
≥ε
)
) ( ( ( 3)
≥ε
)
) ( ( ( 3)
≥ε
)
a
∃ε
/ ∀δ ( 3
<δ ∧
4
∃ε
/ ∀δ ( 3
<δ ∨
c
∃ε
/ ∀δ ( 3
<δ ∧
5
∃ε
/ ∀δ ( 3
≥δ ∧
∃ε
/ ∀δ ( 3
<δ →
)
M3
→ :
3 )
( 3 ∧ : M3 )
27. La c$ntra5ict$ria 5* +S&$ &$" c$n$%i"ta" "$n %ini"tr$" 5 c$n$%ía "* a T$5$" &$" c$n$%i"ta" "$n %ini"tr$" 5 c$n$%ía 4 Nin,n %ini"tr$ 5 c$n$%ía " c$n$%i"ta c A&,)n$" %ini"tr$" 5 c$n$%ía "$n c$n$%i"ta" 5 P$c$" %ini"tr$" 5 c$n$%ía n$ "$n c$n$%i"ta" L$" c$n$%i"ta" n$ "$n %ini"tr$" 5 c$n$%ía
→ A3 )
J3
∃3
( ( ( 3)
≥ε
)
1. N,ar &a 6r$6$"icin* ∃ 3! ∈ ¡ / ( 3 + 7 < ! ) ∧ ( 2 ! − 3 ≥ 3 ) a
∀ 3! ∈ ¡
( 3 − 7 ≥ ! ) ∨ ( 2! + 3 < 3 )
4
∀ 3! ∈ ¡
( 3 + 7 > ! ) ∨ ( 2! − 3 < 3 )
28. +L$" ,i6ci$" n$ "$n n$ a(rican$" ") $4r"a =)ia& a* 1. A&,)n$" ,i6ci$" "$n a(rican$". 2. A&,)n$" a(rican$" "$n ,i6ci$". 7. E" a4")r5$ =) nin,n ,i6ci$ " a(rican$. 8. E" (a&"$ =) nin,n ,i6ci$ n$ " a(rican$. <. N$ " n$ (a&"$ =) =)in=)ira "a n$ a(rican$ a %n$" =) "a ,i6ci$. S$n cirta"* a 7 ! < 4 1 2 ! 8 c 1 2 7 ! < 5 S&$ 1 1 2 ! 7 2<. La c$ntra6)rta t$ta& 5 &a $4r"a 5 &a c$ntra5ict$ria 5 +Cirt$" can5i5at$" "$n tcnic$" "* a T$5$" &$" can5i5at$" "$n tcnic$". 4 A&,)n$" tcnic$" "$n can5i5at$". c Nin,n tcnic$ " can5i5at$. 5 T$5$" &$" can5i5at$" "$n n$ tcnic$". T$5$" &$" tcnic$" "$n n$ can5i5at$". 2>. R&aci$nar &a" 6r$6$"ici$n" ! ")" ti6$".
IDEPUNP / SUPERINTENSIFO 200> I. II. III. IF.
RAHONAMIENTO LGICO
a! c$n5)ct$r" tr%ic$" =) n$ "$n %ta&". Nin,)na 6&a!a 6r)ana " 64&ica. T$5$ 6$&ític$ " i5a&i"ta. La %a!$ría 5 c$nct$r" &,ic$" "$n %$n95ic$".
A. B. C. D.
Unir"a& a(ir%atia Partic)&ar n,atia Partic)&ar a(ir%atia Unir"a& n,atia
a 4 c 5
IB IID IIIC IFA IB IIC IIIA IFD IB IID IIIA IFC IC IID IIIA IFB IA IIC IIID IFB
HOJA DE CLAES
Cic&$ Sti%4r-Dici%4r 2011 C!rso*
RAHONAMIENTO LOGICO.
Se"ana:
0
Te"a * Pr$6$"ici$n" Cat,rica"
Preg!nta
C#a$e
Tie"po %Min&'
Di(ic!#tad
01 02 07 08 0< 0> 0? 0 0 10 11 12 17 18 1< 1> 1? 1 1 20 21 22 27 28 2< 2>
C E A D B A C E C D D E E B D A C E B C A A D E E C
2 2 2 2 7 7 7 7 7 7 7 7 7 2 2 2 7 2 7 7 7 7 2 7 7 2
# # # # M M M M M M M M M # # # M # M M M M # M M #
