# Integer Addition is Commutative/Proof 2

## Theorem

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

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

## Proof

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

Then:

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

$\blacksquare$