by ph-quiett
Last Updated August 14, 2019 00:20 AM

I've been trying to solve this problem but I am not quite sure if I understood it correctly. The question states that:

Assume for all of the question that $P$ is a predicate defined on natural numbers and:

$\forall n \in N: P(n) \implies P(n+1)$

Part A asks: Assume $P(165)$, state whether the following is True, False or could be either:

$P(164)$ -- I put True

$P(166)$ -- True

Part B asks: Assume $\neg$ $P(165)$, state whether the following is True, False or could be either:

$P(164)$ -- False

$P(166)$ -- Either

Part C asks: state whether the following is True, False or could be either:

$P(165) \vee \exists n \in N: n < 165 \wedge \neg P(n) $

And I wasn't quite sure how to tackle C.

Lastly, Part D: Assume for this part that also $P(0)$ and prove or disprove without using induction:

$\exists k \in N: \neg P(k) \wedge \forall n \in N: n < k \implies P(n)$

Part A:

What if $P(n)$ stands for "$n<165$"? And what if it stands for "$n<164$"?

Part B seems ok

Part $C$: reat it loud: "Property $P(n)$ is true for $n=165$, or there exists some number $n$ smaller than $165$ such that $P(n)$ is false". You can think in cases: either $P(165)$ or $\lnot P(165)$. What does Part B tell you?

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