**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.
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