Divides is Antisymmetric

Theorem

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

Corollary

Let $a, b \in \Z$.

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

Proof

We have $\forall a, b \in \Z: a \backslash b \land b \backslash a \implies \left\vert{a}\right\vert = \left\vert{b}\right\vert$ which follows from Integer Absolute Value Greater than Divisors:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle a \backslash b\)	\(\implies\)	\(\displaystyle \left\vert{a}\right\vert \le \left\vert{b}\right\vert\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Integer Absolute Value Greater than Divisors
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle b \backslash a\)	\(\implies\)	\(\displaystyle \left\vert{b}\right\vert \le \left\vert{a}\right\vert\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Integer Absolute Value Greater than Divisors
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \left\vert{a}\right\vert = \left\vert{b}\right\vert\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

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

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

Hence the result.

$\blacksquare$

Proof of Corollary

If $a \backslash b$ and $b \backslash a$ then from the main result $\left\vert{a}\right\vert = \left\vert{b}\right\vert$.

The result follows from Every Integer Divides Its Negative and Every Integer Divides Its Absolute Value.

$\blacksquare$

Sources

• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 3.10$: Theorem $17 \ \text{(ii)}$