Integer Subtraction is Closed

Theorem

The set of integers is closed under subtraction:

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

Proof

From the definition of subtraction:

$a - b := a + \paren {-b}$

where $-b$ is the inverse for integer addition.

From Integers under Addition form Abelian Group, the algebraic structure $\struct {\Z, +}$ is a group.

Thus:

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

Therefore integer subtraction is closed.

$\blacksquare$

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 2$. Operations: Example $1$
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: The Real Number System: $2$
• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers