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 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