# Predicate logic implication

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:

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

Tags :

#### Answers 1

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?