We have $\varphi = a_1 \wedge a_2 \wedge \cdots \wedge a_n$ with n $\in \mathbb{N}$.
$\Psi$ is a set that contains formulas that are implied by $\varphi$. For $\varphi = a\land b$ we could take $\Psi =\{a, b, \neg a \vee b\}$. I want to find the biggest possible $\Psi$.
My idea is $2^{2^n -1}$. We can try to build a truth table, we have n variables, so we have $2^n$ rows. $\varphi$ is only $1$ if all variables are $1$. $0 \to 1$ and $0 \to 0$ is true but only $1 \to 1$ is true. This means that one row has to be filled with $1$. Now we will try to create as many columns as possible. The other rows can have a $0$ or a $1$, so we can create $2^{2^n -1}$ columns.
Does this make sense? Or is it just $2^{2n -1}$? I'm a bit confused.