Express each of these sentences in terms of $Q(x, y)$, quantifiers, and logical connectives, Let Q(x,y) be the statement “x has been a contestant on quiz show y”, where the
domain of x is the set of students and the domain for y consists of all
quiz shows. For each of the English sentences below, please express it in terms of
$Q(x,y)$ with quantifiers.
Please Correct me:
(i) Alice has never appeared in Jeopardy. $\quad\exists x \exists y \neg Q(x,y) $
(ii) Every quiz show has had a student as a contestant. $\quad \forall y \exists x Q(x,y)$
(iii) No student has appeared in both Wheel of Fortune and Family Feud:


*

*$\neg \exists x_1\neg \exists x_2 Q((x_1, \text{Wheel of Fortune}) \land \exists x_2 Q(x_2, \text{Family Feud}))$

 A: *

*Let $Q(x,y)$ be the statement “$x$ has been a contestant on quiz show $y$”, 


*

*the domain of $x$ is the set of students and 

*the domain for $y$ consists of all quiz shows.


(i) Alice has never appeared in Jeopardy.



*

*Let $a$: Alice (assuming Alice is a student). 

*Let $j$: jeopardy.

*Then we have,  $\lnot Q(a, j)\tag{i}$

*Given (i), it is certainly true that your translation "$\exists x \exists y \neg Q(x,y)$" follows from (i): "There is some student x and some quiz show y such that $Q(x, y)$".  


*

*But your translation is not a translation of the given sentence:  Your translation says nothing to the effect that "Alice has never been a contestant of jeopardy". 

*A quantifier is not appropriate* here. We need only the predicate $Q(x, y)$, where "x" is replaced by a named constant for "Alice", and "y" is replaced by a named constant for "Jeopardy". 




(ii) Every quiz show has had a student as a contestant.



*

*Yes, you're correct:  $\forall y \exists x Q(x,y)\tag{ii}$Nice work!



(iii) No student has appeared in both Wheel of Fortune and Family Feud.



*

*We need only one variable to represent a student, and we need only one quantifier: 
Here, we want to say something like: 


*

*"There does not exist a student x (or there is no student x) such that (x has appeared in Wheel of Fortune AND x has appeared in Family Feud)". 

*Equivalently, we can state "for all students x, it is not the case that (x has appeared in Wheel of Fortune AND x has appeared in Family Feud)." 


*To simplify matters, let's let


*

*$f: $ Family feud

*$w: $ Wheel of Fortune



$\neg\exists x (Q(x, w) \land Q(x, f))\quad\equiv\quad \forall x \lnot(Q(x, w) \land Q(x, f))\tag{iii}$


*

*Can you see why your answer for (iii) is problematic? Try translating it into natural language and see if it matches the original sentence.

A: Hints.
(i) is plainly wrong -- you need a constant denoting Jeopardy rather than the second quantifier.
(iii) is plainly wrong -- what you have written implies no one has ever appeared in Wheel of Fortune. You only need a single quantifier.
