Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

266 MATHEMATICAL PHILOSOPHY
stituent, we have xiRyi, but, in general, not yiRxi; that
the domain of R is the class of all the x's,-the class of all
the terms,-such that each of them has the relation to
something or other; and that the codomain of R is the
class of all the y's,-the class of all the terms,-such that,
given any one of them, something or other has the relation
to it. I may add that the terms in the domain of R are
often called the referents of R and that the terms in the
codomain are called the relata of R. Some relations have
fields; others, not. R has a field if and only if the domain
and codomain are of the same type,-that is, are composed of individuals or else of classes of individuals or
else of classes of classes of individuals, and so on,-and
the field is, if there be one, the logical sum ofthe domain
and codomain,-the class, that is, containing every term
in the domain or in the codomain and no other term.
Thus, if O(x, y) be-x is a husband of y-then, if yi be a
wife of xi, the couple (xi, yi) is a constituent of the relation; the relation " husband of" is the class of all such
couples; the domain is the class of husbands; the codomain is the class of wives; the field is the class of husbands
and wives; each husband and nothing else is a referent;
each wife and nothing else is a relatum; observe that in
this example, the domain and codomain have no common
terms. If p(x, y) be-x is a positive integer less than a
positive integer y-then the relation is the class of all
couples (xi, yi) such that Xi and yl are positive integers of
which the former is the less; every integer is a referent;
every integer except I is a relatum; and so, you see, the
domain includes the codomain, but the converse is not
true. If R were identity, for example, or equality or
diversity, then, as you easily see, the domain and the
codomain would each include the other-they would