Suppose $P(n)$ is a predicate. Find the integers $n$ such that $P(n)$ is true.

Question

$P(0)$ is true; for all positive integers $n$, if $P(n)$ is true, then $P(n+2)$ is true.

So my understanding is that $n$ can be $0, 2, 4, 6, 8,...$ (all positive even integers bigger or equal to zero).

Since

$P(0)\implies P(2)$

$P(2)\implies P(4)$

And so on.

However, the answer to this question is that $n$ is $0$ only.

Why is this so?

score 2 · Answer 1 · answered Dec 15 '21 at 06:39

2

$0$ is not a positive integer, so the induction rule cannot be applied at all from $P(0)$ being true to derive the truth of higher $P(n)$.

answered Dec 15 '21 at 06:39

Parcly Taxel

Why does it have to be a positive integer in order for the induction rule to be applied? – FarmerZee Dec 15 '21 at 06:40
@FarmerZee It is stated right there. You cannot apply the induction rule when $n=0$. – Parcly Taxel Dec 15 '21 at 06:41

1 Answers1