I think the following formula is not possible, right? How can I come from the left side to the right? Is there a rule for this transformation I am not aware of or is there an error in my solution?
$$(\neg X \land Y) \lor \neg Z \Leftrightarrow (\neg X \lor \neg Z) \land (\neg Y \lor Z)$$
To apply the distributive rule it should result in: $$(\neg X \land Y) \lor \neg Z \Leftrightarrow (\neg X \lor \neg Z) \land ( Y \lor \neg Z)$$
Thanks in advance.
On
Remember the distributive laws $$p\vee (q\wedge r)\equiv (p\vee q)\wedge(p\vee r)$$
$$p\wedge (q\vee r)\equiv (p\wedge q)\vee(p\wedge r)$$
The first formula which you have is not possible as per the laws of distribution.
The second formula which you have is correct. For the second law we can compare it to the distributive law. When we compare we get, $$p=\neg Z, q=\neg X, r=Y$$ $$\neg Z\vee(\neg X\wedge Y)\equiv(\neg Z \vee\neg X)\wedge(\neg Z \vee Y)$$
The second formula is indeed correct. The first one was mixing up the negations in developing the second clause.
To analyze it: For the second clause in your solution to evaluate to true, no matter what truth-value $Z$ takes, at least one of $\{X,Y\}$ has to be assigned false. Knowing this, let true be assigned to $X$, $Y$ and $Z$. Then the right-hand side cannot evaluate to true, but the left-hand side does. So they certainly cannot be truth-value-equivalent.
Furthermore, knowing this, there can be no deduction from the left- to the right-hand side, by whatever logical means, since deductions preserve the truth value of a statement, i.e. true statements remain true under reformulation by logical rules. By the above mentioned analysis, if there was such a deduction of the LHS of your solution to the RHS, this rule would be violated in your case.