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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:

• 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

• \(x\cdot z=y\cdot z\) and
• \(z \cdot x=z \cdot y\)

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

Table of Contents

Proofs: 1

1. Proposition: Contraposition of Cancellative Law for Multiplying Integers

Mentioned in:

Proofs: 1 2 3 4 5
Propositions: 6

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Footnotes