Error in Introduction to Mathematical Philosophy Is this an error in the text or am I reading incorrectly. What am I missing?
Introduction to Mathematical Philosophy Page 18 Definition of Number
“A relation is said to be “one-one” when, if $x$ has the relation in question to $y$, no other term $x_0$ has the same relation to $y$, and $x$ does not have the same relation to any term $y_0$ other than $y$. When only the first of these two conditions is fulfilled, the relation is called “one-many”; when only the second is fulfilled, it is called “many-one.” It should be observed that the number $1$ is not used in these definitions.”
Error 1:
First condition: … no other term $x’$ has the same relation to $y$. When only the first of these two conditions is fulfilled, the relation is called “one-many”;
Given that $x$ has the relation to $y$ then $x$ is the domain of the relation. Therefore if $x’$ has a relation to $y$, two elements of the domain would map to a single element in the co-domain. This is a many-one. The book refers to this as a one-many.
Error 2:
Second condition: … and $x$ does not have the same relation to any term $y’$ other than $y$. When only the second is fulfilled, it is called “many-one.”
Given that $x$ has the relation to $y$ then $x$ is the domain of the relation. Therefore if $x$ has a relation to $y’$ then $x$ maps to two elements of the co-domain this is a one-many. The book refers to this as a many-one.

It might help to consider the examples that Russell gives in the very next paragraph:

In Christian countries, the relation of husband to wife is  one-one ;
  in Mahometan countries it is one-many ; in Tibet  it is many-one. The
  relation of father to son is one-many ;  that of son to father is
  many-one, but that of eldest son to father  is one-one.

E.g., an x-y relation is one-many if one x can have many y's (but each y has only one x).
