Proposition: Addition of Rational Numbers Is Cancellative
The addition of rational numbers is cancellative, i.e. for all rational numbers \(x,y,z\in\mathbb Q\), the following laws (both) are fulfilled:
Left cancellation property:
If the equation \(z + x=z + y\) holds, then it implies \(x=y\).
Right cancellation property:
If the equation \(x + z=y + z\) holds, then it implies \(x=y\).
Conversely, the equation \(x=y\) implies
- \(x+z=y+z\) and
- \(z + x=z + y\)
for all \(x,y,z\in\mathbb Q\).
Table of Contents
- Proposition: Contraposition of Cancellative Law for Adding Rational Numbers
Proofs: 1 2 3
- Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013