# Divisor Relation is Antisymmetric

## Theorem

Divides is a antisymmetric relation on $\Z_{>0}$, the set of positive integers.

That is:

$\forall a, b \in \Z_{>0}: a \divides b \land b \divides a \implies a = b$

### Corollary

Let $a, b \in \Z$.

If $a \divides b$ and $b \divides a$ then $a = b$ or $a = -b$.

## Proof

Let $a, b \in \Z_{> 0}$ such that $a \divides b$ and $b \divides a$.

Then:

 $\displaystyle a \divides b$ $\implies$ $\displaystyle \size a \le \size b$ Absolute Value of Integer is not less than Divisors $\displaystyle b \divides a$ $\implies$ $\displaystyle \size b \le \size a$ Absolute Value of Integer is not less than Divisors $\displaystyle$ $\leadsto$ $\displaystyle \size a = \size b$

If we restrict ourselves to the domain of positive integers, we can see:

$\forall a, b \in \Z_{>0}: a \divides b \land b \divides a \implies a = b$

$\blacksquare$