# Usuario:JULIAN.D.OR/ejercicio20

Utilice las leyes de la tabla 2.4 y la logica de proposiciones para

a) demostrar que ${\displaystyle \forall _{x}}$P( x ) ${\displaystyle \land }$ ${\displaystyle \forall _{y}}$( P( y ) ${\displaystyle \vee }$ Q( y ) ) ${\displaystyle \equiv }$ ${\displaystyle \lnot }$${\displaystyle \exists _{x}}$${\displaystyle \lnot }$P( x )

b) demostrar la validez de ${\displaystyle \lnot }$${\displaystyle \exists _{x}}$( ${\displaystyle \exists _{y}}$( P( y )${\displaystyle \land }$ Q( y ) ) ${\displaystyle \Rightarrow }$ R( x ) ) ${\displaystyle \Rightarrow }$ ${\displaystyle \forall _{x}}$${\displaystyle \forall _{y}}$( ${\displaystyle \lnot }$P( y ) ${\displaystyle \vee }$ ${\displaystyle \lnot }$Q( y ) ${\displaystyle \vee }$ R( x ) ).

a) ${\displaystyle \forall _{x}}$P( x ) ${\displaystyle \land }$ ${\displaystyle \forall _{y}}$( P( y ) ${\displaystyle \vee }$ Q( y ) ) ${\displaystyle \equiv }$ ${\displaystyle \forall _{x}}$P( x ) ${\displaystyle \land }$ ${\displaystyle \forall _{y}}$( P( y ) ${\displaystyle \vee }$ Q( y ) ) Principio universal de identidad

${\displaystyle \equiv }$ ${\displaystyle \forall _{x}}$P( x ) Ley de simplificacion
${\displaystyle \equiv }$ ${\displaystyle \lnot }$( ${\displaystyle \lnot }$${\displaystyle \forall _{x}}$P( x ) ) doble negacion
${\displaystyle \equiv }$  ${\displaystyle \lnot }$${\displaystyle \exists _{x}}$${\displaystyle \lnot }$P( x ) negacion del universal

b) ${\displaystyle \lnot }$${\displaystyle \exists _{x}}$( ${\displaystyle \exists _{y}}$( P( y )${\displaystyle \land }$ Q( y ) ) ${\displaystyle \Rightarrow }$ R( x ) ) ${\displaystyle \Rightarrow }$ ${\displaystyle \forall _{x}}$${\displaystyle \lnot }$( ${\displaystyle \exists _{y}}$( P( y )${\displaystyle \land }$ Q( y ) ) ${\displaystyle \Rightarrow }$ R( x ) ) Negacion del existencial

${\displaystyle \Rightarrow }$ ${\displaystyle \forall _{x}}$${\displaystyle \lnot }$( ${\displaystyle \lnot }$${\displaystyle \exists _{y}}$( P( y )${\displaystyle \land }$ Q( y ) ) ${\displaystyle \vee }$ R( x ) )  Eliminacion de la implicacion

${\displaystyle \Rightarrow }$ ${\displaystyle \forall _{x}}$${\displaystyle \lnot }$( ${\displaystyle \forall _{y}}$${\displaystyle \lnot }$( P( y )${\displaystyle \land }$ Q( y ) ) ${\displaystyle \vee }$ R( x ) ) negaciion del existencial
${\displaystyle \Rightarrow }$ ${\displaystyle \forall _{x}}$${\displaystyle \forall _{y}}$( ${\displaystyle \lnot }$P( y ) ${\displaystyle \vee }$ ${\displaystyle \lnot }$Q( y ) ${\displaystyle \vee }$ R( x ) ) leyes de De morgan