## 4.6 Relationships between Number Domains

In the previous sections we have introduced the sets **N**, **Z**, **Q**, **R**,
and **C** with the property
**N** subset ^{0} **Z**, **Z** subset ^{1} **Q**, **Q** subset ^{2} **R**, **R** subset ^{3} **C**

where the relations subset ^{i} denote the corresponding embeddings
that we will discuss in Section *Embedding Sets*. These
embeddings preserve all relevant notions (functions and predicates) and the
corresponding properties such that we can operate in the domains as if we
actually had the relationship
**N** subset **Z**, **Z** subset **Q**, **Q** subset **R**, **R** subset **C**.

Indeed there exists for every domain `D` an "identical twin"
`D`' which is a subset of the next larger domain:

We will from now on operate with these twins pretending that they are the
siblings that we have actually defined.
Consequently, we do not any more bother whether + denotes +_{Z} or
+_{Q}; they have essentially the same properties with respect to
computing and reasoning. Of course, we must still take care of that some
functions have no result in a particular domain (the difference of two natural
numbers is not necessarily a natural number) but only in an enclosing domain
(the difference of two natural numbers is always an integer number).

Author: Wolfgang Schreiner

Last Modification: October 4, 1999