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
Famous quotes containing the words proof of and/or proof:
“The source of Pyrrhonism comes from failing to distinguish between a demonstration, a proof and a probability. A demonstration supposes that the contradictory idea is impossible; a proof of fact is where all the reasons lead to belief, without there being any pretext for doubt; a probability is where the reasons for belief are stronger than those for doubting.”
—Andrew Michael Ramsay (1686–1743)
“A short letter to a distant friend is, in my opinion, an insult like that of a slight bow or cursory salutation—a proof of unwillingness to do much, even where there is a necessity of doing something.”
—Samuel Johnson (1709–1784)