Proof of Associativity
We prove associativity by first fixing natural numbers a and b and applying induction on the natural number c.
For the base case c = 0,
Each equation follows by definition ; the first with a + b, the second with b.
Now, for the induction. We assume the induction hypothesis, namely we assume that for some natural number c,
Then it follows,
- (a + b) + S(c)
- = S((a + b) + c)
- = S(a + (b + c))
- = a + S(b + c)
- = a + (b + S(c))
In other words, the induction hypothesis holds for S(c). Therefore, the induction on c is complete.
Read more about this topic: Proofs Involving The Addition Of Natural Numbers
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