Math Help - natural numbers...

1. natural numbers...

THere is a proof that integers can be constructed from the natural numbers
by identifying Z witht he quotient set NxN/~.
Can anyone help with this proof??

2. Re: natural numbers...

You can define a relation $R$ on $\mathbb{N}\times\mathbb{N}$ by $(a,b)R(c,d)\Leftrightarrow a+d=c+b$.
Prove that $R$ is an equivalence relation (how?).
The equivalence classes for $R$ now form a group, which is isomorphic to the integers, with the (class independent- prove it for fun) operation $[(a,b)]+[(c,d)]=[(a+b,c+d)]$.

3. Re: natural numbers...

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.