Rational Subtraction is Closed

Theorem

The set of rational numbers is closed under subtraction:

$\forall a, b \in \Q: a - b \in \Q$

Proof

From the definition of subtraction:

$a - b := a + \paren {-b}$

where $-b$ is the inverse for rational number addition.

From Rational Numbers under Addition form Abelian Group, $\struct {\Q, +}$ forms a group.

Thus:

$\forall a, b \in \Q: a + \paren {-b} \in \Q$

Therefore rational number subtraction is closed.

$\blacksquare$