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

According to the cancellation law for adding real numbers, we have for all real numbers \(x,y,a\in\mathbb R\): \[\begin{array}{rcl}z + x=z + y&\Longleftrightarrow &x=y,\\ x + z=y + z&\Longleftrightarrow& x=y. \end{array}\] 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 real numbers \(x\) and \(y\) are unequal, then the inequality is preserved if we add an arbitrary real 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