You can define a relation on by .
Prove that is an equivalence relation (how?).
The equivalence classes for now form a group, which is isomorphic to the integers, with the (class independent- prove it for fun) operation .
You can define a relation on by .
Prove that is an equivalence relation (how?).
The equivalence classes for now form a group, which is isomorphic to the integers, with the (class independent- prove it for fun) operation .
Note that we can think of the "natural numbers" as a subset of the integers, defined in this way, by associating the natural number n with the class containing the pair (a, a+ n). The number "0" is then the class containing (n, n) for all natural numbers n.