Ordering of Integers is Reversed by Negation

Theorem

Let $x, y \in \Z$ such that $x > y$.

Then:

$-x < -y$

Proof

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

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

We have:

\(\ds x\)

\(>\)

\(\ds y\)

\(\ds \leadsto \ \ \)

\(\ds \eqclass {a, b} {}\)

\(>\)

\(\ds \eqclass {c, d} {}\)

Definition of Integer

\(\ds \leadsto \ \ \)

\(\ds a + d\)

\(>\)

\(\ds b + c\)

Definition of Strict Ordering on Integers

\(\ds \leadsto \ \ \)

\(\ds b + c\)

\(<\)

\(\ds a + d\)

\(\ds \leadsto \ \ \)

\(\ds \eqclass {b, a} {}\)

\(<\)

\(\ds \eqclass {d, c} {}\)

\(\ds \leadsto \ \ \)

\(\ds -\eqclass {a, b} {}\)

\(<\)

\(\ds -\eqclass {c, d} {}\)

Negative of Integer

\(\ds \leadsto \ \ \)

\(\ds -x\)

\(<\)

\(\ds -y\)

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: Exercise $1$