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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.

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  • $\begingroup$ I think you are correct. I may be missing something just like you, though. $\endgroup$ – 5xum Mar 20 '15 at 13:15
  • $\begingroup$ See Deepak's answer: "no other term" means your $x'$ does not exist. Conversely, if $x$ and $x'$ both map to $y$ then the first condition is false, because there is another term in the same relation to $y$. $\endgroup$ – David K Mar 20 '15 at 13:32
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You are misreading it completely. The first condition is that no other $x_0$ has the same relation to $y$. Means that if this condition is fulfilled, you cannot have $x$ and a distinct $x_0$ mapping to a single $y$. Clearly, this cannot be many ($x$) to one ($y$). However, since a single $x$ can map to multiple $y$ values, this is a one to many mapping.

The same applies to the other statement. Try re-reading it carefully.

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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).

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