The multiplication of natural numbers is cancellative, if the number we want to "cancel out" does not equal \(0\)1, i.e. for all \(x,y,z\in\mathbb N\) with \(z\neq 0\) the following laws (both) are fulfilled:
Left cancellation property: If the equation \(z \cdot x=z \cdot y\) holds, then it implies \(x=y\).
Right cancellation property: If the equation \(x \cdot z=y \cdot z\) holds, then it implies \(x=y\).
Conversely, the equation \(x=y\) implies
for all \(x,y,z\in\mathbb N\) with \(z\neq 0\).
Proofs: 1
Please note that if we allow \(z=0\), then the proposition is wrong: e.g. \(0\cdot 5=0\cdot 3\), but \(5\neq 3\). ↩
