# Need help performing a tree method to test for satisfiability

For those who commented on my previous questions, sorry for the lack of information and explanation. Clearly I did not do a good job of explaining myself so I deleted the question and hope this one goes better.

EDIT: here is what I have so far, can anyone tell me if it is right/wrong? http://www.box.net/shared/static/hrv9rh7ukt.jpg

I know how to perform standard tree method satisfiable tests but this question I am having trouble with because it has to do with elections.

Question:

Use the tree method to determine whether or not the following set of sentences is statisfialbe. If so, specify a satisfying case for its atomic sentences (ie aRjb, dRic, bRja, and cPid).

{bRja $\rightarrow$ dRic, dRic $\rightarrow$ $\neg$cPid, dRic $\rightarrow$ aRjb, $\neg$dRic $\rightarrow$ cPid, cPid $\rightarrow$ bRja, aRjb $\rightarrow$ cPid}

I think the format of the sentences is standard nomenclature but if it is not, ill will do my best to explain it. P refers to Social Preternce while R refers to Rule.

UPDATE:

I set the following

A = aRjb

B = bRja

C = cPid

D = dRic

So then the tree looks like:

• $B \lor D$
• $D \lor \neg C$
• $D \lor A$
• $D \lor C$
• $C \lor B$
• $A \lor C$

Maybe its because I haven't done even standard tree methods in a while but can anyone help me on how to check if it is satisfiable?

EDIT UPDATE:

bPia $\rightarrow$ cPia

aPic $\rightarrow$ (aPib $\lor$ bPic)

Conc, ($\neg$aPib $\land$ $\neg$bPia) $\land$ ($\neg$aPic $\land$ $\neg$cPia) $\rightarrow$ ($\neg$aPic $\land$ $\neg$cPia)

Can I solve it like the rest by substituting each "clause" with a letter?

So I would set bPia to B1, bPic to B2, cPia to B1, aPib to A1, aPic to A2, and solve like a normal tree?

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Hi eric -- glad to see you haven't given up yet :-) Thanks for making more of an effort this time to explain what you're talking about. I don't know the format of the sentences (which may or may not indicate that it's not "standard nomenclature" :-), but I don't think it's relevant, since they're referred to as "atomic" in the question, so if I understand correctly this is just a logic question and one doesn't have to know about any of the election stuff to answer it? – joriki Apr 5 '11 at 6:20
Thanks, @"joriki. I believe this is simply a logic question however I think the format would be considered an election style because cPid means i prefers c to d. – user9128 Apr 5 '11 at 6:24
@eric: You says you know how to perform standard tree-method satisfiability tests, and you agree that this is simply a logic question. Then you should explain what's preventing you from applying the standard method that you know. – joriki Apr 5 '11 at 6:30
@eric: In case the problem is that these sentences are not in the form of disjunctive clauses, note that $A \rightarrow B$ is equivalent to $\neg A \lor B$. – joriki Apr 5 '11 at 6:32
well i'm not 100% sure it is purely a logic question. I don't see how I would apply the normal logic rules to such format as cPid, which means i would prefer c to d. – user9128 Apr 5 '11 at 6:33

• $B \lor D$
• $D \lor \neg C$
• $D \lor A$
• $D \lor C$
• $C \lor B$
• $A \lor C$
that making $A, B, C, D$ all true is a satisfying assignment.