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

TRANSFORMATION 157
class, of C', as by the law (9): (a -> a'), (b -> a'), (c -> b'),
or into the whole of it, as by the law (io): (a - a'),
(b -> a'), (c -> b'), (c - c'). You will note that neither
(9) nor (io) is a one-to-one transformation. And by a
little work, or a little play, you will readily discover that
the possibilities of transformation increase or decrease
with the sizes of the classes and vary with other circumstances such as whether the classes are of the same size
(containing the same number of elements), whether the
classes have elements in common,-overlap or intersect,
as we say,-possibly coincide, and especially whether the
classes are finite (like those we have considered) or
infinite (like the class, for example, of all the integers,
l, 2, 3,... and so on endlessly).
Let us think a bit about the effect of such circumstances.
Suppose C' has more elements than C has; it is plain
that no one-to-one transformation of C into C' can cover
the whole of C', but it can be covered by transformations
that are not one-to-one.
Suppose a of C is the same as a of C',-C and C'
thus intersecting; then in the above transformations
(I),..., (6), C itself contains the transform of one of
its own elements, a being the transform of a in (I) and
(2), of b in (3) and (5) and ofc in (4) and (6).
Suppose C and C' coincide,-the elements of either
being those of the other,-an important special case of
intersection; then, you see, each of the transformations
(I),..., (8) converts (transforms) the class C into itself.
Next suppose C to be composed of all the integers
I, 2, 3,... n,...; and suppose C' to be composed of all
the even integers 2, 4, 6..., 2n,...; you notice that C'
is a part, or subclass, of C; now let us transform C by the