Solution of the 1st Homework
نویسنده
چکیده
Solution. We prove that n + m is a natural number using mathematical induction on n. • (Base case) When n = 0, we have 0 + m = m ∈ N for all m ∈ N by definition of addition. • (Induction step) Suppose that n + m ∈ N for all m ∈ N. What we want to prove is that (n++) + m ∈ N for all m ∈ N. But since n + m ∈ N by the induction hypothesis, (PA2) shows that (n++) + m = (n + m)++ ∈ N as desired. This proves the induction step. Therefore by (PA5), the mathematical induction, it follows that n + m is a natural number for all n,m ∈ N.
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