ONT Re: Relations And Their Divisitudes
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RATD. Note 3
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Forming an ordered pair involves a 3-adic relation, much in the same way that
forming the sum of two integers involves the 3-adic relation that is known as
the "additive group of the integers" G = (J, +) c J x J x J, where J is the
set of integers {..., -3, -2, -1, 0, 1, 2, 3, ...}.
Let's take a space X that is big enough to include all the entities you want.
The operation cons is a total function cons : X x X -> X
The operation car is a partial function car : X -> X
The operation cdr us a partial function cdr : X -> X
Note that cons is a 3-adic relation, albeit of a special type,
specifically, one that is functional on its first two doamins.
These three operations are given an operational definition
by the following equations, for applicable values of z.
car (cons (x, y)) = x
cdr (cons (x, y)) = y
cons (car z, cdr z) = z
x y
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z
In short, when you form an ordered pair z = cons (x, y) there are
three things being related, the elements x and y and the pair z.
This 3-adicity of the pairing operation is not just some trick,
but comes to have a critical bearing, both in the formalization
of relations in axiomatic set theory, and further in the actual
implementation of computational systems.
Jon Awbrey
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