Integer Addition is Commutative

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

$\blacksquare$

Sources

• 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 5$: The system of integers
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers: $\mathbf Z. \, 2$
• 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): $\S 1$