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

According to the cancellation law for multiplying integers, we have for all integers \(x,y,z\in\mathbb Z\) 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: \[x \neq y\Longleftrightarrow \begin{cases} z \cdot x\neq z \cdot y,&\text{or}\\ x \cdot z\neq y \cdot z, \end{cases}\]

i.e. if any two integers \(x\) and \(y\) are unequal, then the inequality is preserved, if we multiply both sides of the inequality by an arbitrary integer \(z\neq 0\). Conversely, if we have an inequality, in which both sides are multiplied by the same integer \(z\neq 0\), then we can "cancel it out", and still preserve the inequality.

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