Induction and Statements

Question

I'm having trouble with induction in my discrete math course.

We are given a statement we know (∀k)(P(k)⇒P(k+2)), where k is an element of N.

After that we have a series of statements, I'll give one for example, that we must say is true, false, or possibly true.

1. ∀n, P(n), where n is and element of N.

I don't even know where to begin with that statement. Also, I know how induction works, but when I just see the implication, it doesn't help me out.

Thank you for reading this, I hope someone can point me in the right direction to solving these.

Answer 1

Induction says that if you know $\forall k, P(k) \implies P(k+1)$ and $P(0)$, then $P(n) \forall n$. So if you only have the first half of the conditions, then the conclusion holds if and only if $P(0)$.

Answer 2

I don't even know where to begin with that statement. Also, I know how induction works, but when I just see the implication, it doesn't help me out.

Exactly.   You don't have enough information from the iteration step alone to either verify or falsify the statement.   You are missing the required base step.

That's your answer.   It's possibly true; iff $P(0)$ and $P(1)$ also holds then $\big\langle\forall n\in \Bbb N: P(n)\big\rangle$.

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$$\overbrace{\;\\[2ex]\underbrace{\big\langle\forall n\in \Bbb N: \big(P(n)\to P(n+2)\big)\big\rangle}_\text{this is what you have} \wedge \underbrace{ P(0)\wedge P(1)\quad}_{\text{but you also need this}} \vdash \underbrace{\big\langle\forall n\in \Bbb N: P(n)\big\rangle}_{\text{in order to show this}}}^\text{Induction} $$