Natural Numbers under Multiplication form Semigroup

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\N$ be the set of natural numbers.

Let $\times$ denote the operation of multiplication on $\N$.


The structure $\struct {\N, \times}$ forms a semigroup.


Proof

Closure

We have that Natural Number Multiplication is Closed.

That is, $\struct {\N, \times}$ is closed.

$\Box$


Associativity

We have that Natural Number Multiplication is Associative.

$\Box$


Thus the criteria are fulfilled for $\struct {\N, \times}$ to form a semigroup.

$\blacksquare$


Sources