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

TRANSFORMATION 169
we happen to have identity, or coincidence, of domain
and codomain, of transform class and transformed class,
of function-range and argument-range; but this is, in general, not so; and 1 recommend that you do the matching
by some other propositional function, say, x is the husband
of y, or x is the specific gravity of y, or the integer x is
greater than the integer y, or x is ethically so sublime that
he should not allow y to make a glass of beer or "turn
water into wine."
1 can not refrain from tarrying here long enough to
illustrate, by just one example, the now evident façt that
any problem, process or operation having to do with
(ordinary) functions is a problem, process or operation
having to do with relations or with transformations, and
conversely. The example is as follows: If R1 and R2
be two relations such that it is significant (true or false
and not merely nonsensical) to say that Ri's codomain
and R2's domain intersect, then there is a relation R' ---
called the relative product of Ri by R2-such that, if
xRîy and yR2z, then xR'z; respecting functions the corresponding fact is this-if F1 and F2 be two functions such
that it is significant to say that the range of Fi's argument
intersects the range of F2, then there is a function F'which might be called the functional product of F1 by F2 -such that, if x =Fi(y) and y =F2(z), then =F'(x);
finally, as to transformations, the corresponding fact is
this-if Ti and T2 be two transformations such that it is
significant to say that the class of transforms (under T4)
and the class of things transformed by T2 intersect,
then there is a transformation T'-called the product of
Ti by T2-such that, if Ti converts x into y and T2
converts y into z, then T' converts x into z. I hope that
what I have now said is sufficient to make clear the exceed