relacions

R es simétrica <==> R = R^{o(-1)}

Si ( R simétrica & R es transitiva ) ==> R es reflexiva.

<x,y> € R

<x,y> € R^{o(-1)}

<y,x> € R

( <x,y> € R & <y,x> € R ) & ( <y,x> € R & <x,y> € R )

<x,x> € R & <y,y> € R

( RoS )^{o(-1)} = ( S^{o(-1)}oR^{o(-1)} )

Si ( R es simétrica & S es simétrica ) ==> RoS = ( SoR )^{o(-1)}

<x,y> € RoS

<x,z> € S & <z,y> € R

<y,z> € R & <z,x> € S

<y,x> € SoR

<x,y> € ( SoR )^{o(-1)}

R es transitiva <==> RoR [<< R

[==>]

<x,y> € RoR

<x,z> € R & <z,y> € R

<x,y> € RoR 

[<==]

<x,y> € R & <y,z> € R

<x,z> € RoR

<x,z> € R 

Si R es transitiva ==> RoR es transitiva

<x,y> € RoR & <y,z> € RoR

( <x,u> € R & <u,y> € R ) & ( <y,v> € R & <v,z> € R )

<x,y> € R & <y,z> € R

<x,z> € RoR

( A [W] B )oR = (AoR) [W] (BoR)

<x,y> € ( A [W] B )oR

<x,z> € A [W] B & <z,y> € R

( <x,z> € A & <z,y> € R ) || ( <x,z> € B & <z,y> € R )

<x,y> € (AoR) [W] (BoR)

( A [M] B )oR = (AoR) [M] (BoR)

<x,y> € ( A [M] B )oR

<x,z> € A [M] B & <z,y> € R [ ( a & a ) <==> a ]

( <x,z> € A & <z,y> € R ) & ( <x,z> € B & <z,y> € R )

<x,y> € (AoR) [M] (BoR)

Si R es transitiva ==> ( R es asimétrica <==> R es irreflexiva )

[==>]

( Si <x,y> € R ==> ¬( <y,x> € R ) )

Sigui y = x ==>

Si <x,x> € R ==>

¬( <x,x> € R )

[<==]

Si ¬( Si <x,y> € R ==> ¬( <y,x> € R ) )

<x,y> € R & <y,x> € R

<x,x> € R