How to Prove the these claims in Propositional Logic?

Question

This Question included multiple sub-questions, I removed those I solved and posted only those I failed (they aren't related).

$\Sigma$ is called special if for each $\alpha, \beta$ in $WFF$: $\Sigma \models \alpha \vee \beta$ or $\Sigma \models \alpha \vee (\lnot \beta)$

Q: Prove or give a counter example:

a) If $\Sigma$ is special then for every 2 assignments $z_1 \not = z_2$: $z_1 \not \models \Sigma$ or $z_2 \not \models \Sigma$

b) $\Sigma$ is special iff for every $\phi$ in $WFF$: $\Sigma \models \phi$ or $\Sigma \models \lnot\phi$

I wasn't able to prove any of these 2 or find a counter example, any help?

I found out that $\{p_i:i\}$ is special, while $\{(p_0 ∧ \lnot p_0)\}$ isn't.

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References: https://en.wikipedia.org/wiki/Well-formed_formula (WFF)

Note: $\Sigma$ refers to a group of elements in $WFF$

Answer 1

for a): This is true. Let's assume the opposite. For a special $\Sigma$, there are $z_1\neq z_2$ such that $z_1 \vDash \Sigma$ and $z_2 \vDash \Sigma$.

So there exists an atom $p_i$ such that $z_1(p_i)\neq z_2(p_i)$. Let's say, without loss of generality, that $z_1(p_i)=0$ and $z_2(p_i)=1.$

Now look at $(p_i\wedge \neg p_i),\; p_i \in WFF$. Does the condition of speciality apply to those expressions? No; I'll leave it up to you to figure out why.

So we've got to a contradiction.

לא ריחמו עלינו עם הגיליון הזה ):