91Ó°ÊÓ

Check which of the following sets are consistent. (a) \(\left\\{\neg p_{1} \wedge p_{2} \rightarrow p_{0}, p_{1} \rightarrow\left(\neg p_{1} \rightarrow p_{2}\right), p_{0} \leftrightarrow \neg p_{2}\right\\}\), (b) \(\left\\{p_{0} \rightarrow p_{1}, p_{1} \rightarrow p_{2}, p_{2} \rightarrow p_{3}, p_{3} \rightarrow \neg p_{0}\right\\}\), (c) \(\left\\{p_{0} \rightarrow p_{1}, p_{0} \wedge p_{2} \rightarrow p_{1} \wedge p_{3}, p_{0} \wedge p_{2} \wedge p_{4} \rightarrow p_{1} \wedge p_{3} \wedge p_{5}, \ldots\right\\}\)

Show that \(\\{\neg\\}\) is not a functionally complete set of connectives. Idem for \(\\{\rightarrow, \vee\\}\) (hint: show that each formula \(\varphi\) with only \(\rightarrow\) and \(\vee\) there is a valuation \(v\) such that \(\llbracket \varphi \rrbracket=1\) ).