(related to Proposition: Contraposition of Cancellative Law for Multiplying Rational Numbers)

According to the cancellation law for multiplying rational numbers, we have for all rational numbers \(x,y,a\in\mathbb Q\) with \(z\neq 0\)

\[\begin{array}{rcl}z \cdot x=z \cdot y&\Longleftrightarrow &x=y,\\ x \cdot z=y \cdot z&\Longleftrightarrow& x=y. \end{array}\] By virtue of the proving principle by contraposition, it follows: \[x \neq y\Longleftrightarrow \begin{cases} z \cdot x\neq z \cdot y,\\ x \cdot z\neq y \cdot z, \end{cases}\]

i.e. if any two rational numbers are unequal, then the inequality is preserved, if we multiply both sides of the inequality by an arbitrary rational number \(a\neq 0\).

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  1. Landau, Edmund: "Grundlagen der Analysis", Heldermann Verlag, 2008