Problem 50 (difficulty: 5/10)

\(\displaystyle A_1,A_2,\ldots\) are logical statements. What can we say about their truth value if

(a) \(\displaystyle A_1\wedge\forall n\in\N~A_n\Rightarrow A_{n+1}\)?

(b) If \(\displaystyle A_1\wedge\forall n\in\N~A_n\Rightarrow(A_{n+1}\wedge A_{n+2})\)?

(c) If \(\displaystyle A_1\wedge\forall n\in\N~(A_n\vee A_{n+1})\Rightarrow A_{n+2}\)?

(d) If \(\displaystyle \forall n\in N ~ \neg A_n\Rightarrow\exists k\in\{1,2,\ldots,n-1\} ~ \neg A_k\)?

Give me another random problem!