## Monday, November 4, 2013

### On Reduction, 10.5 of 11

Carrollian Logistic Identities

This is an addendum to last Friday’s blog about logistics. Here are some logistic identities:
(x+y) ~ (x <+> y)    =    (x~y) + ( x <+> 1/x <+> y <+> 1/y )
(x <+> (y~z) )  +  (y <+> (z~x) )  +  (z <+> (x~y) )      =
1/(x<+>x<+>y<+>y<+>z<+>z) <+> (M(x,y,z)+M(x,y,z))

(x + (y<+>z)) <+> (y + (z<+>x)) <+>  (z + (x<+>y))    =
M(x,y,z) <+> M(x,y,z)

(x <+> (y+z)) + (y <+> (z+x)) +  (z <+> (x+y))             =
M(x,y,z)  + M(x,y,z)

M(x,y,y)   =   ( ( x <+> y ) + y )  <+> (x + x + y )

M(x,y,y)   =   ( ( x + y ) <+> y )  + (x <+> x <+> y )

When interpreted as boolean logic equations, you get these identities:

(x and y) iff (x or y)     equals
(x iff y) and (x or not x or y or not y)

(x or (y iff z)) and (y or (z iff x)) and (z or (x iff y))   equals
not (x or x or y or y or z or z) or ((most of x,y,z) and (most of x,y,z))

(x and (y or z)) or (y and (z or x)) or (z and (x or y))   equals
(most of x,y,z) or (most of x,y,z)

(x or (y and z)) and (y or (z and x)) and (z or (x and y))   equals
(most of x,y,z) and (most of x,y,z)

Most of x,y,y    equals    ( (x or y) and y )  or ( x and x and y )
Most of x,y,y    equals    ( (x and y) or y )  and ( x or x or y )
Because of their surrealistic sound, I propose that we call these identities “Carrollian”.