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

SECTION C.
ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS.
Summary of Section C.
In the present section we have to consider three very important classes of
relations, of which the use in arithmetic is constant. A one-many relation is
a relation R such that, if y is any member of C('R, there is one, and only one,
term x which has the relation R to y, i.e. R'y e 1. Thus the relation of father
to son is one-many, because every son has one father and no more. The
relation of husband to wife is one-many except in countries which practise
polyandry. (It is one-many in monogamous as well as in polygamous countries,
because, according to the definition, nothing is fixed as to the number of relata
for a given referent, and there may be only one relatum for each given referent
without the relation ceasing to be one-many according to the definition.) The
relation in algebra of x2 to x is one-many, but that of x to x2 is not, because
there are two different values of x that give the same value of x2.
When a relation R is one-many, R'y exists whenever ye (1'], and vice
versa; i.e. we have
R e one-many. _: y e ('R. Dy. E! R'y.
Thus relations which give descriptive functions that are existent wlenever
their arguments belong to the converse domains of the relations in question
-- 4-v
are one-many relations. Hence Cnv, D, (1, C, R, R, sg, gs, Re, ), s, 8, l, I, l,
Cl, R1 are all of them one-many relations.
When R is a one-many relation, R'y is a one-valued function; conversely,
every one-valued function is derivable from a one-many relation. A manyvalued function of y is a member of R'y, where I'y is not a unit class, and
any one of its Ilembers is regarded as a value of the function for the argument y; but a one-valued function of y is the single term R'y which is
obtained when R is one-many. Thus for example the sine would, in our
notation, appear as a relation, i.e. we should put
sin = [ = y-y3/3! + yS/5!-...} Df,
whence sin'y = y - y:/3! + y/5!-...,