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Proof

(related to Proposition: Algebraic Structure of Rational Numbers Together with Addition)

The set of rational numbers \(\mathbb Q\), together with the specific addition operation "\(+\)" is a commutative group. This is because:

1. The addition operation is associative, i.e. \((x+y)+z=x+(y+z)\) is valid for all \(x,y,z\in\mathbb Q\).
2. We have shown the existence of a neutral element of addition - the number \(0\in\mathbb Q\), i.e. such that \(0+x=x\) for all \(x\in\mathbb Q\).
3. For every \(x\in\mathbb Q\), there there exists an inverse rational number \(-x\in\mathbb Q\), such that \(x+(-x)=0\).
4. For every \(x\in\mathbb Q\), there there exists an inverse rational number \(-x\in\mathbb Q\), such that \(x+(-x)=0\).
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References

Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013