Natural Number Subtraction is not Closed

Theorem

The operation of subtraction on the natural numbers is not closed.

Proof

By definition of natural number subtraction:

$n - m = p$

where $p \in \N$ such that $n = m + p$.

However, when $m > n$ there exists no $p \in \N$ such that $n = m + p$.

$\blacksquare$

Sources

• 1937: Richard Courant: Differential and Integral Calculus: Volume $\text { I }$ (2nd ed.) ... (previous) ... (next): Chapter $\text I$: Introduction: $1$. The Continuum of Numbers: $1$. The System of Rational Numbers and the Need for its Extension
• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.1$. Subsets closed to an operation: Example $88$
• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 2$. Operations: Example $1$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 27$. Binary operations