Integer Addition is Associative

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Theorem

The operation of addition on the set of integers $\Z$ is associative:

$\forall x, y, z \in \Z: x + \left({y + z}\right) = \left({x + y}\right) + z$


Proof 1

From the formal definition of integers, $\left[\!\left[{a, b}\right]\!\right]$ is an equivalence class of ordered pairs of natural numbers.

From Integers under Addition form Abelian Group, the integers under addition form a group, from which associativity follows a priori.

$\blacksquare$


Proof 2

Let $x = \left[\!\left[{a, b}\right]\!\right]$, $y = \left[\!\left[{c, d}\right]\!\right]$ and $z = \left[\!\left[{e, f}\right]\!\right]$ for some $x, y, z \in \Z$.


Then:

\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle x + \left({y + z}\right)\) \(=\) \(\displaystyle \) \(\displaystyle \left[\!\left[{a, b}\right]\!\right] + \left({\left[\!\left[{c, d}\right]\!\right] + \left[\!\left[{e, f}\right]\!\right]}\right)\) \(\displaystyle \) \(\displaystyle \)          by formal definition of integers          
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\displaystyle \left[\!\left[{a, b}\right]\!\right] + \left[\!\left[{c + e, d + f}\right]\!\right]\) \(\displaystyle \) \(\displaystyle \)          by definition of integer addition          
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\displaystyle \left[\!\left[{a + \left({c + e}\right), b + \left({d + f}\right)}\right]\!\right]\) \(\displaystyle \) \(\displaystyle \)          by definition of integer addition          
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\displaystyle \left[\!\left[{\left({a + c}\right) + e, \left({b + d}\right) + f}\right]\!\right]\) \(\displaystyle \) \(\displaystyle \)          Natural Number Addition is Associative          
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\displaystyle \left[\!\left[{a + c, b + d}\right]\!\right] + \left[\!\left[{e, f}\right]\!\right]\) \(\displaystyle \) \(\displaystyle \)          by definition of integer addition          
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\displaystyle \left({\left[\!\left[{a, b}\right]\!\right] + \left[\!\left[{c, d}\right]\!\right]}\right) + \left[\!\left[{e, f}\right]\!\right]\) \(\displaystyle \) \(\displaystyle \)          by definition of integer addition          
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\displaystyle \left({x + y}\right) + z\) \(\displaystyle \) \(\displaystyle \)          by formal definition of integers          

$\blacksquare$


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