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:
\(\ds a \divides b\) | \(\implies\) | \(\ds \size a \le \size b\) | Absolute Value of Integer is not less than Divisors | |||||||||||
\(\ds b \divides a\) | \(\implies\) | \(\ds \size b \le \size a\) | Absolute Value of Integer is not less than Divisors | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \size a = \size b\) |
If we restrict ourselves to the domain of positive integers, we can see:
- $\forall a, b \in \Z_{>0}: a \divides b \land b \divides a \implies a = b$
$\blacksquare$