I need to proof that
$$P[\varphi] \le P[\psi]\text{ does not imply }P[\varphi\rightarrow\psi] = 1 $$
$$P[\psi] + \varepsilon ≥ P[\varphi]\text{ does not imply }P[\varphi\rightarrow\psi] \ge 1− \varepsilon$$
$$P[\varphi] = P[\psi]\text{ does not imply }P[\varphi\leftrightarrow\psi] = 1 $$
I have the following information
$$ \operatorname{Pr}[\varphi]=\sum_{\beta \in Q,\ \beta | \models \varphi} p(\beta) $$
Any advice?
Actual state:
For the first I have at the moment
$$P(A) = P(B)$$
$$P(A|B) = P(B|A)$$
$$\frac{P(A \wedge B)}{P(B)}=\frac{P(A \wedge B)}{P(A)}$$