Expressing given statements using quantifiers examples

Question

I'm new on this subject and I have answered some questions which I found. Since there are no answers for them; I couldn't be sure that my answers are true.

Could you help me verify the answers or help me where I made mistakes?

Given functions: W(x,z) = "x likes to watch z tv series", K(x,y) = "x and y are coworkers".

"Everyone likes to watch exactly one tv series": $ \forall x (W(x,z) \land \forall k (W(x,k) \rightarrow (k=z)) $

"There is a person that all of his coworkers are also coworker with each other": $ \exists x \forall y (K(x,y) \land \forall k(K(y,k) \rightarrow (y \not= k))) $

"There are at least two person who like to watch Simpsons tv series": $ \neg [\forall x \forall y (W(x,Simpsons) \land W(y,Simpsons) \rightarrow (x=y)) ] $

and also this is a reverse question;

"$ \forall x (K(x,John) \rightarrow (W(x,The Walking Dead) \leftrightarrow (x = Frank))) $": Everyone is coworker of John if and only if his name is Frank and he likes to watch The Walking Dead tv series.

Thanks for your guidance.

Answer 1

Your answers are not quite right. First off, usually when one writes something like "$\forall x$", a set has to be given from which you take every $x$; i.e. "$\forall x \in \mathbb{R}$". Of course here you're not dealing with precise mathematical objects but I think it would still make more sense to specify the origin of your variables. Also, every variable you use has to be specified beforehand (you didn't do so in the first example); and it's usually tidier if you put all the parts containing quantifiers first and the implications and actual statements second.

So here's how I'd rewrite the first one:

$$(\exists z \in \{\mbox{Series}\} )(\forall x \in \{\mbox{People}\})(\forall k \in \{\mbox{Series}\}):(W(x,z) \wedge (W(x, k)) \implies (z=k))$$

Equivalently, one could write

$$(\exists z \in \{\mbox{Series}\} )(\forall x \in \{\mbox{People}\})(\forall k \in \{\mbox{Series}\}):(W(x,z) \wedge (k\not=z \implies \neg(W(x,k)))$$

Hints for the others:

• For the second one your statement says "There is a person $x$ who is a coworker of everyone else and for every pair of people $k,y$, $k$ being a coworker of $y$ implies that $k \not= y$". What you're trying to say is "There is a person $x$ such that for all pairs of people $k$ and $y$, $k$ being a coworker of $x$ and $y$ being a coworker of $x$ implies that $k$ is a coworker of $y$" (in this formulation basically you just have to substitute quantifiers and mathematical notation in place of the words).
• The third one is correct. (you might consider trying to write it without the negation though - there's actually a simple rule that tells you how to negate statements with quantifiers, you should see if you can come up with it)
• For the fourth notice that the equivalence sign is not between $K(x, \mbox{"John"})$ and the rest, but between $W(x, \mbox{"The walking dead"})$ and $x = \mbox{"Frank"}$. The correct answer will be something starting with "If $x$ is a coworker of John, then...".

I hope that's at least somewhat clear. :)