On Logistic, 3 of 8; Einstein Addition

         3. Einstein Addition

Now consider the “equivalence” connective

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

         =  (x and y) or (not x and not y)

They have these arithmetical counterparts:

    (x {+} 1/y) + (y {+} 1/x) 

    (x + y) {+} (1/x + 1/y) 

These expressions are identical! They equal:

    (x+y) / (1+xy)

          Which also equals:

(1{+}xy) / (x{+}y)

Call this operator “equivalence”, or “Einstein addition”, after relativistic velocity addition. Denote it as (x~y). It has these laws:

(tanh x)~(tanh y)  =  tanh(x+y)

x~0           =       x

x~(1/0)           =      1/x

x~1           =       1

x~ -1         =      -1

x~ -x         =       0

x~ -1/x       =       1/0

1~ -1         =       %

-(x~y)        =     (-x)~(-y)

1/(x~y)       =     (1/x)~y     =   x~(1/y)

1/(x~y)       =      (x {+} y) + (1/x {+} 1/y) 

                                        =     (x + 1/y) {+} (1/x + y) 

                                        =     (1+xy) / (x+y)

(1/x)~(1/y)   =      x~y

x~y           =      y~x

(x~y)~z       =    x~(y~z)

(x~y)~z        =   (x+y+z+xyz) / (xy+yz+zx+1)

                                         =   (x{+}y{+}z{+}xyz) / (xy{+}yz{+}zx{+}1)

These imply Transposition:

(A~x) = B    if and only if   x = (-A~B)

Note that multiplication and equivalence relate to negative and reciprocal in opposite ways:

-(x*y) = (-x)*y = -1*x*y  ;   1/(x*y) = (1/x)*(1/y)

-(x~y) =  (-x)~(-y)       ;   1/(x~y) = (1/x)~y =  (1/0)~x~y

Negation distributes into multiplication and over equivalence; whereas reciprocal distributes over multiplication and into equivalence. To multiplication, negation is a term (-1) and reciprocal is a sign; to equivalence, reciprocal is a term (1/0) and negation is a sign.