Proofs Involving The Addition of Natural Numbers - Definitions

Definitions

This article will use the definitions in addition of natural numbers, particularly and :

  • a + 0 = a
  • a + S(b) = S(a + b)

For the proof of commutativity, it is useful to define another natural number closely related to the successor function, namely "1". We define 1 to be the successor of 0, in other words,

.

Note that for all natural numbers a,

\begin{align} S(a) &= S(a + 0)\\ &= a + S(0)\\ &= a + 1 \end{align}

due to and .

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