## Theorem

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

$\forall x, y \in \Z: x + y = y + x$

## Proof 1

From the formal definition of integers, $\eqclass {a, b} {}$ is an equivalence class of ordered pairs of natural numbers.

From Integers under Addition form Abelian Group, the integers under addition form an abelian group, from which commutativity follows a fortiori.

$\blacksquare$

## Proof 2

Let $x = \eqclass {a, b} {}$ and $y = \eqclass {c, d} {}$ for some $x, y \in \Z$.

Then:

 $\ds x + y$ $=$ $\ds \eqclass {a, b} {} + \eqclass {c, d} {}$ Definition of Integer $\ds$ $=$ $\ds \eqclass {a + c, b + d} {}$ Definition of Integer Addition $\ds$ $=$ $\ds \eqclass {c + a, d + b} {}$ Natural Number Addition is Commutative $\ds$ $=$ $\ds \eqclass {c, d} {} + \eqclass {a, b} {}$ Definition of Integer Addition $\ds$ $=$ $\ds y + x$ Definition of Integer

$\blacksquare$