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

According to the cancellation law for adding rational numbers, we have for all rational numbers \(x,y,a\in\mathbb Q\): \[\begin{array}{rcl}z + x=z + y&\Longleftrightarrow &x=y,\\ x + z=y + z&\Longleftrightarrow& x=y. \end{array}\] By virtue of the proving principle by contraposition, it follows \[x \neq y\Longleftrightarrow \begin{cases} z + x\neq z + y,&\text{or}\\ x + z\neq y + z, \end{cases}\]

i.e. if any two rational numbers \(x\) and \(y\) are unequal, then the inequality is preserved if we add an arbitrary rational number \(z\) to both sides of the inequality. Conversely, given an inequality, in which the same rational number \(z\) is added on both sides, we can "cancel it out" and preserve the inequality.

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