On Logistic, 4 of 8; the Mediant

4. The Mediant

          Define the “mediant” M(x,y,z) thus:

          M(x,y,z)       =

         

(x + (y{+}z)) {+} (y + (z{+}x)) {+}  (z + (x{+}y))   

                                                     +

(x + (y{+}z)) {+} (y + (z{+}x)) {+}  (z + (x{+}y))   

          In logistic, this translates to:

(x and (y or z)) or (y and (z or x)) or  (z and (x or y))   

                                                    and

(x and (y or z)) or (y and (z or x)) or  (z and (x or y))   

-        a sing-song version of the majority operator.

          Mediant equals:

                    (xy+yz+zx) / (x+y+z)

          It also equals:

                    (xy{+}yz{+}zx) / (x{+}y{+}z)

         

          This is also an identity:

          M(x,y,z)       =

                   

(x {+} (y+z)) + (y {+} (z+x)) + (z {+} (x+y)) 

                                                  {+}

                    (x {+} (y+z)) + (y {+} (z+x)) + (z {+} (x+y)) 

          Mediant has these laws:

                    M(a, -a, a) = -a

                    M(b, -b, a) = -b²/a

          Geometric Mean:

                    M(1 , b , b²)     =   b

                    M(a , ab , ab²)  =  ab

                    M(a² , ab , b²)   =  ab

                    M(a, root(ab) , b)   =  root(ab)

          Recall:

                    M(a, a, a)  = a

          Multiplication distributes:

                    k*(M(x,y,z))  =  M( kx , ky , kz )

          Left Division Distributes:

                    k/(M(x,y,z))  = M( k/x , k/y , k/z )

          Additions:

                    M(x,0,y)     =      x {+} y

                    M(x,∞,y)    =        x + y

          Triple Product:

                    M(x,y,z)*(x+y+z)*(x{+}y{+}z)  =  xyz

          Cancellation:

                    M(x, y, M(-x, -y, z))   =   z

          Transposition:

                    M(x,y,z) = w    if and only if   z  =  M(-x,-y,w)

          Swapping:

                    M(x,y,z) = w    if and only if   M(x,y,-w) = -z

          Hyperbolic Associativity:

                    M(x, y, M(a,b,c))  =   M(a, b, M(x,y,c))   if and only if  xy = ab