Natural Number Addition is Closed

Theorem

The operation of addition on the set of natural numbers $\N$ is closed:

$\forall x, y \in \N: x + y \in \N$


Proof

Follows directly from Natural Numbers under Addition form Commutative Monoid.

A monoid by definition is a semigroup.

Again by definition, the operation in a semigroup is closed.

$\blacksquare$


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 $88$
  • 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 8.9$
  • 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $1$: Numbers: Real Numbers: $1$
  • 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): Chapter $1$: Complex Numbers: The Real Number System: $1$
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