Proposition: Multiplication of Integers Is Cancellative

The multiplication of integers is cancellative, i.e. for all integers \(x,y,z\in\mathbb Z\), with \(z\neq 0\) the following laws (both) are fulfilled1:

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

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

Proofs: 1

  1. Proposition: Contraposition of Cancellative Law for Multiplying Integers

Proofs: 1 2 3 4 5
Propositions: 6

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  1. Note that this proposition would be obviously wrong if we allow \(z\) to equal \(0\), e.g. for \(z=0, x=5, y=3\) we would get \(0\cdot 5=0\cdot 3\), but \(5\neq 3\).