truth table equivalency I am stuck on this question and attempting to answer it makes me feel that its equivalent to searching for a needle in a large pond...
I need help with this, can someone explain how I even attempt to find the solution to this?
Question: Find a logical statement equivalent to $(A \to B) \&  \sim  C$, the statement must use only operators $\sim, |$.
I know that I can do $(A \& \sim B) \, | \, C$ which is logically equivalent but it says not to use anything other than $\sim, |$. The statement I have uses "$\&$".

I will assume that "|" is NAND operator defined as :
$A | B \Leftrightarrow \lnot(A \land B)$
If it is so then we can write :
$(A \rightarrow B) \land \lnot C \Leftrightarrow (\lnot A \lor B) \land \lnot C \Leftrightarrow (\lnot A \land \lnot C) \lor (B \land \lnot C) \Leftrightarrow$
$\Leftrightarrow \lnot(\lnot A \mid \lnot C) \lor \lnot(B \mid \lnot C) \Leftrightarrow \lnot ((\lnot A \mid \lnot C) \land (B \mid \lnot C)) \Leftrightarrow$
$\Leftrightarrow (\lnot A \mid \lnot C) \mid (B \mid \lnot C)$
On the other hand if " | " is OR operator then we have :
$(A \rightarrow B) \land \lnot C \Leftrightarrow (\lnot A \lor B) \land \lnot C \Leftrightarrow \lnot(\lnot(\lnot A \lor B) \lor C)$
