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Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple Eglish. b. "There is a student in this class who has chatted with exactly one other student"
I looked up the answers to the this problem, but I have no friggin clue how they got there.
If this is what I think it is, are we talking about things like "there exists a number x such that" blah blah?
Statement: ∃x∃y( Q(x,y) ∧ ∀z ( z ≠ x ∨ ~Q(x,z))))
given the universe of discourse being defined as all students in this class, and Q(x,y) being defined as x has chatted with y
Did you come up with that statement?
No, and I only kind of understand why z is there.
The z is there to make sure there isn't another person with whom they spoke.
z would be hypothetically the other person, so it's saying such a person does not exist.
namely, when we say one other student we mean that Q(x,x) can be true or false?
but x=y is a possiblity
in which case x has both chatted with x as well as with y, meaning x has chatted with 2 people
or is Q(x,x)=F for all cases?
Technically this statement allows yourself to be the person you're talking with. It doesn't not say \(x\neq y\)
but the main issue is after I negate teh statement I have to translate it into english (which isn't a big deal gettign wrong since I have the answer key), but I have no clue how to do it
You just have a lot of freedom.
so you're saying that based off of the original english (the statement was not part of the problem, but I believe it is part of the standard answer key) you can be the person you have chatted with
Negate it logically first.
meaning that if you have chatted with yourself then you have not chatted with anyone else?
Well that is a possibility.
But you don't necessarily have to note that out. It's a logical consequence of the statement.
because if you have chatted with yourself that's exactly one person you've chatted with
It could be that \(Q(x.x)\) always results in false because it has a \(x\neq y\) baked into it.
can you show me how to construct the statement from scratch again?
In other works, it could be that \(Q(x,y)\) means "\(x\) has talked to \(y\) and \(x\) is not \(y\)"
If you've already been given a statement, stick with the statement you were given. Do you have to make up the statement yourself?
that statement I have to make up
What is the statement do you have to make up?
"∃x∃y( Q(x,y) ∧ ∀z ( z ≠ x ∨ ~Q(x,z))))" that's the first part of the problem -I translate the engilsh sentence into a statement. I looked at a lot of logically equiavlent statements online by searching for the answers to the problem, and this one made the most sense to me
So that is the answer you found online?
sorry it's ∃x∃y( Q(x,y) ∧ ∀z ( z = x ∨ ~Q(x,z))))
the last disjunction was originally given in implication form p -> q but I changed it into ~p V q since it was just all a mess to me
so the original statement I was given was ∃x∃y( Q(x,y) ∧ ∀z ( z ≠ x → ~Q(x,z))))
oh wait http://math.stackexchange.com/questions/73300/negation-of-uniqueness-quantifier
Yeah, you want to build around that.
Use that to make \(y\) unique.
Then around that put ∃x ( [...] ∧ x ≠ y) where [...] is the uniqueness for y.