Integer Addition is Associative

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Theorem

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

$\forall x, y, z \in \Z: x + \paren {y + z} = \paren {x + y} + z$


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 $a, b, c, d, e, f \in \N$ such that:

$x = \eqclass {a, b} {}$, $y = \eqclass {c, d} {}$ and $z = \eqclass {e, f} {}$.


Then:

\(\ds x + \paren {y + z}\) \(=\) \(\ds \eqclass {a, b} {} + \paren {\eqclass {c, d} {} + \eqclass {e, f} {} }\) Definition of Integer
\(\ds \) \(=\) \(\ds \eqclass {a, b} {} + \eqclass {c + e, d + f} {}\) Definition of Integer Addition
\(\ds \) \(=\) \(\ds \eqclass {a + \paren {c + e}, b + \paren {d + f} } {}\) Definition of Integer Addition
\(\ds \) \(=\) \(\ds \eqclass {\paren {a + c} + e, \paren {b + d} + f} {}\) Natural Number Addition is Associative
\(\ds \) \(=\) \(\ds \eqclass {a + c, b + d} {} + \eqclass {e, f} {}\) Definition of Integer Addition
\(\ds \) \(=\) \(\ds \paren {\eqclass {a, b} {} + \eqclass {c, d} {} } + \eqclass {e, f} {}\) Definition of Integer Addition
\(\ds \) \(=\) \(\ds \paren {x + y} + z\) Definition of Integer

$\blacksquare$


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