In this question, we are dealing with predicate logic, where we have access to the deduction theorem, the soundness theorem, and the completeness theorem. The context of this question is the following:
First we say a set ${\Delta}$ of formulae is deductively closed iff ${ \Delta \vdash \sigma}$ implies that ${\sigma \in \Delta}$
So before this question, we have shown certain things to be true in our system mainly:
1- if ${\{ \Delta_i \}}$ are all deductively closed sets of formulae then so is ${\bigcap\limits_i \Delta_i}$
2- Any set ${\Gamma}$ of formulae is contained in a smallest deductively closed set ${\Delta(\Gamma)}$
3- We have established that ${\Delta(\Gamma) = \{ \phi : \Gamma \vdash \phi \}}$
4- We have shown that if ${\Sigma}$ is complete, then it is deductively closed.
(In our system, we have defined a complete set as a set that is consistent and for each sentence $\phi$ in a language $L$, exactly one of $\phi$ and $\neg \phi$ belongs to the set.
Also, a consistent set is a set that is only capable of deriving one of $\phi$ or $\neg \phi$, but not both.)
5- We aslo shown that if ${ \Delta(\Gamma) \cap \{ \phi, \neg \phi\} = \emptyset}$, then ${\Delta(\Gamma) \cup \{\phi \}}$ and ${ \Delta(\Gamma) \cup \{ \neg \phi \} }$ are both consistent.
Furthermore, we say that a set $\Sigma$ is maximal consistent for a language $L$ if $\Sigma$ is consistent and for any consistent set of sentences $\Sigma'$ in $L$ with $\Sigma \subseteq \Sigma'$, we have $\Sigma = \Sigma'$.
So to show the above question these are the only things that we have established and that we can use. I tried solving this question but It doesn't seem to work out especially that my solution was completely off, and I don't know what to do no, please help!!