I'm sorry in advance for any wrong terminology, english is not my first language, so I'll do my best.
I am trying to solve the following problem:
Let M be the minority link of three positions (meaning that $\overline{V}(M \phi\psi\tau)$ is always different from the majority of the values $\overline{V}(\phi), \overline{V}(\psi) \text{ and } \overline{V}(\tau)$. Show that:
- the set $\{M, \mathcal{F}\}$ is adequate (where $\mathcal{F}$ is the link of 0-arguments that for any $V$ takes the value $\overline{V}(\mathcal{F})=\mathcal{F}$.
- $\{M\}$ is inadequate (prove first that if $\phi$ is using only $M$ and $A$, then if in a $V$ we take the opposite of $V(A)$ (note it as $V'$), then $\overline{V'}(\phi)$ will be the opposite of $\overline{V}(\phi)$.
I have started by noting down that $\overline{V}(M\phi\psi\tau)$ being the minority link means that it's equiveland to $(\neg((\phi\land\psi)\lor(\phi\land\tau)\lor(\psi\land\tau)))$. Also, in order to prove both sentences, I need to show that if a propositional formula is using $\mathcal{F}$ it will always take the value $F$, but if it's using $M$, its value will be determined by $M$. I now get stuck as to how to prove this.
Similar to 1, number 2 is weird to me for two reasons: a) how does using $A$ help at all with proving that $M$ alone is inadequate and b) I don't undestand why it shouldn't be adequate.
Can someone explain where I need to begin from and help me understand the proof of this?