Proposition: Multiplication Of Rational Numbers Is Cancellative

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

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

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

Proofs: 1

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

Proofs: 1 2

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  1. Note that this proposition would be obviously wrong if we allow \(z\) to equal \(0\), e.g. for \(z=0, x=\frac 15, y=\frac 23\) we would get \(0\cdot \frac 15=0\cdot \frac 23\), but \(\frac 15\neq \frac 23\).