Natural Number Addition is Closed

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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