Metric for Integer modulo n

Definition

• With ${\mathbb{Z}}_p$ where $p\ne 2$ and $p$ is Prime Number
• Let $d(a,b)=\min \{a{+}_{\mathrm{mod}p}(-b),b{+}_{\mathrm{mod}p}(-a)\}$

Proof of Metric

Showing $d(a,b)$ non-negative

• By defn of ${+}_{\mathrm{mod}p}$, $a{+}_{\mathrm{mod}p}(-b)$ and $b{+}_{\mathrm{mod}p}(-a)$ are both in $\{0,\dots ,p-1\}$, thus, $d(a,b)\ge 0$

Showing $d(a,b)$ is symmetric

• Let $a,b\in {\mathbb{Z}}_p$
• Then, $d(a,b)=\min \{a{+}_{\mathrm{mod}p}(-b),b{+}_{\mathrm{mod}p}(-a)\}$ and $d(b,a)=\min \{b{+}_{\mathrm{mod}p}(-a),a{+}_{\mathrm{mod}p}(-b)\}$
• Both, $d(a,b)$ and $d(b,a)$ minimize the same set, thus they are equal

Showing $d(a,b)=0⟺a=b$

• First observe, $d(a,b)=0$ is equivalent to $a{+}_{\mathrm{mod}p}(-b)=0$ or $b{+}_{\mathrm{mod}p}(-a)=0$
• We know that Integer Modulo p is a Field
• As the additive inverse is unique,
  • $a{+}_{\mathrm{mod}p}(-b)=0⟺-b=-a$
  • $⟺a=b$
• Similarly,
  • $b{+}_{\mathrm{mod}p}(-a)=0⟺-a=-b$
  • $⟺a=b$ Note that we proved both sides, as all conjunctions are Biconditional Statements.

Showing Triangle Inequality