# Usuario:JULIAN.D.OR/ejercicio10

Demuestre lo siguiente por derivacion

a) ${\displaystyle \forall _{x}}$P( x ) ${\displaystyle \vdash }$ ${\displaystyle \forall _{x}}$( P( x ) ${\displaystyle \vee }$ Q( x )

b) ${\displaystyle \exists _{x}}$ ${\displaystyle \forall _{y}}$P( x , y ) ${\displaystyle \vdash }$ ${\displaystyle \forall _{y}}$ ${\displaystyle \exists _{x}}$ P( x , y )

c)${\displaystyle \exists _{x}}$ ${\displaystyle \forall _{y}}$P( x , y , z ) ${\displaystyle \vdash }$ ${\displaystyle \exists _{z}}$ P( z , z , z )

a)${\displaystyle \forall _{x}}$P( x ) ${\displaystyle \vdash }$ ${\displaystyle \forall _{x}}$( P( x ) ${\displaystyle \vee }$ Q( x )

1.${\displaystyle \forall _{x}}$P( x ) premisa

2.P( a ) ${\displaystyle S_{a}^{x}}$ Particularizacion del Universal 1.

3.P( a ) ${\displaystyle \vee }$ Q( x ) ley de combinacion

4.${\displaystyle \forall _{x}}$( P( x ) ${\displaystyle \vee }$ Q( x ) ) Generalizacion del Universal 3.

b)${\displaystyle \exists _{x}}$ ${\displaystyle \forall _{y}}$P( x , y ) ${\displaystyle \vdash }$ ${\displaystyle \forall _{y}}$ ${\displaystyle \exists _{x}}$ P( x , y )

1.${\displaystyle \exists _{x}}$ ${\displaystyle \forall _{y}}$P( x , y ) premisa

2.${\displaystyle \forall _{y}}$P( a , y ) ${\displaystyle S_{a}^{x}}$ Particularizaion del existencial 1.

3.P( a , b ) ${\displaystyle S_{b}^{y}}$ Particularizaion del Universal 2.

4.${\displaystyle \exists _{x}}$P( x , b ) Generalizacion del Existencial en 3.

5.${\displaystyle \forall _{y}}$ ${\displaystyle \exists _{x}}$ P( x , y ) generalizacion del Universal en 4.

c)${\displaystyle \exists _{x}}$ ${\displaystyle \forall _{y}}$P( x , y , z ) ${\displaystyle \vdash }$ ${\displaystyle \exists _{z}}$ P( z , z , z )

1.${\displaystyle \exists _{x}}$ ${\displaystyle \forall _{y}}$P( x , y , z ) premisa

2.${\displaystyle \forall _{y}}$P( z , y , z ) ${\displaystyle S_{z}^{x}}$ Particularizaion del existencial 1.

3.P( z , z , z ) ${\displaystyle S_{z}^{y}}$ Particularizaion del Universal 2.

4.${\displaystyle \exists _{z}}$ P( z , z , z ) Generalizacion del Existencial