# Rational Subtraction is Closed

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## 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.

$\blacksquare$

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $1$: Integral Domains: $\S 2$. Operations: Example $1$