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 Infinite 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

  • 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Introduction
  • 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 2$. Operations: Example $1$
  • 1973: G. Stephenson: Mathematical Methods for Science Students (2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.1$ Real Numbers
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