Proposition: Addition of Integers

Any integer \(x\in\mathbb Z\) is by the corresponding proposition an equivalence class. \[x:=[a,b],\] where \(a\) and \(b\neq 0\) denote natural numbers representing the equivalence class \(x\). Given two integers \(x:=[a,b]\), \(y:=[c,d]\), \(a,b,c,d\in \mathbb N\), the addition of integers is defined by using the addition of the natural numbers \(a+c\) and \(b+d\):

\[\begin{array}{rcl} x+y&:=&[a+c,b+d], \end{array}\]

where \([a+c,b+d]\) is also an integer, called the sum of the integers \(x\) and \(y\). The sum exists and is well-defined, i.e. it does not depend on the specific representatives \([a,b]\) and \([c,d]\) of \(x\) and \(y\).

Proofs: 1

  1. Proposition: Addition of Integers Is Associative
  2. Proposition: Addition of Integers Is Commutative
  3. Proposition: Addition of Integers Is Cancellative
  4. Proposition: Existence of Integer Zero (Neutral Element of Addition of Integers)
  5. Proposition: Uniqueness of Integer Zero
  6. Proposition: Existence of Inverse Integers With Respect to Addition

Definitions: 1 2
Proofs: 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Propositions: 18 19 20 21 22 23 24 25 26 27

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