Proposition: Multiplication of Natural Numbers Is Cancellative

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:

Conversely, the equation \(x=y\) implies

for all \(x,y,z\in\mathbb N\) with \(z\neq 0\).

Proofs: 1

  1. Proposition: Contraposition of Cancellative Law for Multiplying Natural Numbers

Proofs: 1 2 3 4

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  1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013


  1. Please note that if we allow \(z=0\), then the proposition is wrong: e.g. \(0\cdot 5=0\cdot 3\), but \(5\neq 3\).