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??

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- May 21st 2005, 01:10 AMzeenatural 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?? - Dec 27th 2014, 06:10 AMRebesquesRe: natural numbers...
You can define a relation $\displaystyle R$ on $\displaystyle \mathbb{N}\times\mathbb{N}$ by $\displaystyle (a,b)R(c,d)\Leftrightarrow a+d=c+b$.

Prove that $\displaystyle R$ is an equivalence relation (how?).

The equivalence classes for $\displaystyle R$ now form a group, which is isomorphic to the integers, with the (class independent- prove it for fun) operation $\displaystyle [(a,b)]+[(c,d)]=[(a+b,c+d)]$. - Dec 27th 2014, 06:19 AMHallsofIvyRe: 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.