Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

438

PROLEGOMENA TO CARDINAL ARITHMETIC

[PART II

so that "sin'y" has the usual meaning of sin y. Then instead of sin-l x, we
should have sin'x, which would be the class of values of sin- x; and instead
of "y=sin-l," which is a misleading notation because y=sin-' and
4 -z=sin-lx do not imply y=z, we should have yesin'x. Similar remarks
would apply to any of the other functions that occur in analysis.
A relation R is called many-one when, if x is any member of D'R, there
is one, and only one, term y to which x has the relation R, i.e. R x e 1. Thus
many-one relations are the converses of one-many relations. When a relation
R is many-one, IR'x exists whenever x eD'R.
A relation is called one-one when it is both one-many and many-one, or,
what comes to the same, when both it and its converse are one-many. Of the
one-many relations above enumerated, Cnv, sg, gs, I, t, t, C1, R1 are one-one.
Two classes a, f are said to be similar when there is a one-one relation RL
such that D'R =a. aR =/, i.e. when their terms can be connected one to
one, so that no term of either is omitted or repeated. We write "a sm / " for
"a is similar to P3." When two classes are similar, the cardinal numbers of
their terms are the same; it is this fact chiefly that makes one-one relations
of fundamental importance in cardinal arithmetic.
According to the above, a relation is one-many when
ye '. Ay. R'y e l,
i.e. when RI(['? C 1.
Similarly a relation is many-one when
4 -R"D'R C 1,
and a relation is one-one when both conditions are fulfilled. The classes
-— ) 4 —
RC"(IR R")'LR, which appear here, are often important; some of their
properties have already been given in *37'77'771'772'773 and in *53-61 to
*53'641.
It is convenient to regard one-many, many-one and one-one relations as
particular cases of relations which, for some given a and /, have
— ~ 4 -R"('R C a. R"D'R C 3.
A —1 4 -We put a - K I R "' c a. R "DlR C 13} Df.
Hence, without a new definition, "1 -* 1" becomes the class of one-one
relations; also, as will be shown, "1 -e Cls " becomes the class of one-many
relations, and " Cls - 1 " becomes the class of many-one relations. Although
it is chiefly these three special values of a — / that are important, we shall
begin by a general study of classes of relations of the form a -,8.