Everyone knows that the right sort of diagram can be a great aid in rendering complex matters comprehensible, so let’s extract the all too compressed bits of the Relation Theory article that it takes to illuminate Peirce’s 1870 “Logic of Relatives” and use them to fashion what icons we can within the current frame of discussion.
For the immediate present, we may start with dyadic relations and describe the most frequently encountered species of relations and functions in terms of their local and numerical incidence properties.
Let be an arbitrary dyadic relation. The following properties of
can then be defined:
If is tubular at
then
is known as a partial function or a pre-function from
to
frequently signalized by renaming
with an alternate lower case name, say
and writing
Just by way of formalizing the definition:
To illustrate these properties, let us fashion a generic enough example of a dyadic relation, where
and where the bigraph picture of
looks like this:
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If we scan along the dimension from
to
we see that the incidence degrees of the
nodes with the
domain are
in that order.
If we scan along the dimension from
to
we see that the incidence degrees of the
nodes with the
domain are
in that order.
Thus, is not total at either
or
since there are nodes in both
and
having incidence degrees less than
Also, is not tubular at either
or
since there are nodes in both
and
having incidence degrees greater than
Clearly then the relation cannot qualify as a pre-function, much less as a function on either of its relational domains.
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