Relation Theory • Preliminaries
Two definitions of the relation concept are common in the literature. Although it is usually clear in context which definition is being used at a given time, it tends to become less clear as contexts collide, or as discussion moves from one context to another.
The same sort of ambiguity arose in the development of the function concept and it may save a measure of effort to follow the pattern of resolution that worked itself out there.
When we speak of a function 
Let us write 
When it comes to parsing the notation 
One way to resolve the ambiguity is to formalize a distinction between the function
Another tactic treats the whole notation 
In categorical and computational contexts, at least initially, the type is regarded as an essential attribute or integral part of the function itself. In other contexts we may wish to use a more abstract concept of function, treating a function as a mathematical object capable of being viewed under many different types.
Following the pattern of the functional case, let the notation 
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-adic or 
and 
for, which takes us back to the question of what we are using our logical systems for.
where 
is the so-called boolean domain 
But we are really using 
or 
my favorite colors for painting the areas of a venn diagram.
is called a “characteristic function” in set theory since it characterizes a subset 
of 
where the value of 
is 
But I like the language they use in statistics, where 
is called an “indicator function” since it indicates a subset of 
and defined as the function 
if and only if 
I like this because it links up nicely with the sense of indication in the calculus of indications.
Other names for it are the “fiber” or “pre-image“ of 
in the rather ugly but pre-eminently useful way as 
Inquiry Into Inquiry 