Predicate Logic

I'm studying for an exam, and I'm not really sure how to portray this:

The domain is all people.

V (w) = w is a voter

P (w) = w is a politician

K (y, z) = y knows z

T (y, z) = y trusts z

Cal is a voter who knows everyone. (Cal is c)

Would this be: ∀x V(c)^K(c,x)

There is a politician that no other polit开发者_如何学Goician trusts

∃x∀y P(x)^P(y)^T(y,x)

I'm not sure if those are right. Wouldn't the last one be saying: There are politicians that no one trusts? How do I make it singular?

Also: No one trusts every politician.

∃x∀y P(y)^T(¬x,y)

Thanks

P.S. I'm not sure if this is posted in the right place, but I assume this would be a good place for it.

The first one is good, you are saying:

"For all persons, cal is a voter and cal knows that person"

(This could be written another way, by the way. You could write it as "Cal is a voter and for all persons, cal knows that person. Would seem a bit more natural and closer to the original plain english statement, but they mean the same thing)

The second one, you are off. You are saying

"There exists a person x such that for all persons y, x is a politician AND y is a politician AND y trusts x"

Try this:

∃x∀y:P(x)^(P(y)->(!T(y,x))

(Sorry, I didn't use all of the proper symbols D:. -> means implies, and ! means not) so this is saying "There exists a person x such that for all persons y, x is a politicion AND if y is a polition, y does not trust x"

For the third one, you want to go for "For all persons, there exists a politician that they do not trust".

First: good

Second: Good, but it should be "not T", right?

Third: "not x" means nothing. You should use DeMorgan's to rephrase as "For every perspn there is an politician whom they don't like".