Integer Addition is Associative

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, $\eqclass {a, b} {}$ is an equivalence class of ordered pairs of natural numbers.

From Integers under Addition form Abelian Group, the integers under addition form a group, from which associativity follows from Group Axiom $\text G 1$: Associativity.

$\blacksquare$

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

$\blacksquare$

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: $\mathbf Z. \, 1$
• 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (next): $\S 1$