Práctica 4 (pre 2010, Paradigmas)

Ejercicio 01

• i) {□}
• ii) {□}
• iii) {□}
• iv) {□}
• v) { {~q,p} }
• vi) { {p},{q,r} }

Ejercicio 02

• i. i-iv Tautologias
• ii. (¬p -> q)^(p -> q)^(¬p -> ¬q) = (pvq)^(~pvq)^(pv¬q) = { {p,q},{~p,q},{p,~q} } = { {p,q},{~p,q},{p,~q},{p} } = { {p,q},{~p,q},{p,~q},{p},{q} } -> Con lo cual debe cumplirse p^q.

Ejercicio 04

• i) $\forall x.\forall y.(Q(x,y)\vee \neg P(x,y))$
• ii) $\forall x.\forall y.(\neg P(x,y)\vee \neg Q(x,y)\vee R(x,y))$
• iii) $\forall x.\exists y.(\neg P(x,y)\vee Q(x,y))$

Ejercicio 05

Por favor ante cualquier error corrijan:

i)
Skolem:
$\forall x.\exists y.x<y$
$=\forall x.x<f(x)$
Clausal: {{x < f(x)}}

ii)
Skolem:
$\exists x.\exists y.x<y$
$=\exists y.a<y=a<b$
Clausal: {{a < b}}

iii)
Skolem:
$\forall x.(\neg P(x)\vee \exists y.(P(y)\wedge Q(y)))$
$=\forall x.(\neg P(x)\vee (P(f(x))\wedge Q(f(x)))))$
$=\forall x.([\neg P(x)\vee P(f(x))]\wedge [\neg P(x)\vee Q(f(x))])$
$=\forall x.[\neg P(x)\vee P(f(x))]\wedge \forall x.[\neg P(x)\vee Q(f(x))]$
Clausal: {{~P(x) v P(f(x))}, {~P(x) v Q(f(x))}}

iv)
Skolem:
$\exists x.\forall y.(P(x,y)\wedge Q(x)\wedge \neg R(y))$
$=\forall y.(P(a,y)\wedge Q(a)\wedge \neg R(y))$
Clausal: {{P(a,y)},{Q(a)},{~R(y)}}

v)
$\forall x.(P(x)\wedge \exists y.(Q(y)\vee \forall z.\exists w.(P(z)\wedge \neg Q(w))))$
$=\forall x.(P(x)\wedge \exists y.\forall z.\exists w.(Q(y)\vee (P(z)\wedge \neg Q(w))))$
$=\forall x.(P(x)\wedge \forall z.(Q(f(x))\vee (P(z)\wedge \neg Q(g(z)))))$
$=\forall x.(P(x)\wedge \forall z.([Q(f(x))\vee P(z)]\wedge [Q(f(x))\vee \neg Q(g(z))]))$
$=\forall x.\forall z.(P(x)\wedge [Q(f(x))\vee P(z)]\wedge [Q(f(x))\vee \neg Q(g(z))])$
Clausal: {{P(x)},{Q(f(x)) v P(z)},{Q(f(x)) v ~Q(g(z))}}

Ejercicio 06

• i)

$\forall c.((C(c)\wedge \exists x.(N(x)\wedge x\in c))->\exists y.(y\in c\wedge \forall z.(z\in c->y<=z)))$

• ii)

$\forall d.((D(d)\wedge \forall c.(P(d,c)\wedge V(c))->F(d)))$
$\forall d.((D(d)\wedge VE(d))->V(d))$
$\forall d.(((D(d)\wedge \exists p.P(p,d)\wedge VE(p))->VE(d))\wedge ((D(d)\wedge \neg \exists p.P(p,d)\wedge VE(p))->R(d)))$