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.