Proof
(related to Proposition: Addition of Integers Is Commutative)
- By definition of integers, the integers $x,y\in Z$ are ordered pairs $x=[a,b],y=[c,d]$ for some natural numbers \(a,b,c,d\in\mathbb N\).
- Therefore, $x+y=[a,b]+[c,d].$
- $=[a+c,b+d],$ by definition of adding integers.
- $=[c+a,d+b],$ due to commutativity law for natural numbers.
- $=[c,d]+[a,b],$ by definition of adding integers.
- $=y+x,$ by definition of integers.
∎
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References
Bibliography
- Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013
