Divisor Relation is Antisymmetric

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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:

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

$\blacksquare$