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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 "$\&$".

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    @PatrickDaSilva $\sim$ is used to represent logical negation in *Principia Mathematica* among many other works. It is a stylized letter 'N'.2012-07-24

3 Answers 3

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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 $[\sim ( (\sim A) | B)] \, | \, C$.

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    I will need to think deeper into this for it to make sense but what u have said is helping me understand this better.2012-01-16
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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 $\sim((A | \sim B) | (\sim C))$

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    But $\neg A$ is the same as $A | A$, so you can eliminate the negation in this sentence.2012-07-24
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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 :

$(A \rightarrow B) \land \lnot C \Leftrightarrow (\lnot A \lor B) \land \lnot C \Leftrightarrow \lnot(\lnot(\lnot A \lor B) \lor C)$

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    Eliminate negation by writing $\neg A$ as $(A | A)$2012-07-24