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 "$\&$".
 A: If
$\sim$ means "non-" 
$A | B$ means "$A$ or $B$" (non-exclusive) 
$A \,\&\, B$ means "$A$ and $B$" 
Then note that we can replace a $A \& B$ by $\sim( (\sim A) | (\sim B))$ because if $A$ and $B$ are true, then the $|$ gives false, and then the $\sim$ in front of it gives you true, but if either $A$ or $B$ is false (assume it's $A$), then $\sim A$ is true, hence $|$ is true and $\sim$ of $|$ gives you false. I hope I was clear enough.
This means that $( A \& (\sim B)) | C$ is equivalent to $[\sim((\sim A) | (\sim (\sim B))] \, | \, C$ which in turn is equivalent to $[\sim ( (\sim A) | B)] \, | \, C$. 
A: It helps me to think about venn diagrams, like here:
http://en.wikipedia.org/wiki/Logical_connective
(I assume | means "alternative denial" like on wikipedia)
So $\alpha \& \beta$ is $\sim (\alpha | \beta)$, and $\alpha \rightarrow \beta$ is $\sim (\alpha \& \sim \beta)$, so using these in combination should work. 
In particular, what you want should be the same thing as $$\sim((A | \sim B) | (\sim C))$$
A: 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 :
$(A \rightarrow B) \land \lnot C \Leftrightarrow (\lnot A \lor B) \land \lnot C \Leftrightarrow \lnot(\lnot(\lnot A \lor B) \lor C)$
