Integer Addition is Commutative

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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 priori.

$\blacksquare$


Proof 2

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

$\blacksquare$


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