## Wednesday, January 8, 2014

### 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.