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:
- $\neg \exists x_1\neg \exists x_2 Q((x_1, \text{Wheel of Fortune}) \land \exists x_2 Q(x_2, \text{Family Feud}))$
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.
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 $Q(x, y)$".
We need only one variable to represent a student, and we need only one quantifier:
Here, we want to say something like:
To simplify matters, let's let
$\neg\exists x (Q(x, w) \land Q(x, f))\quad\equiv\quad \forall x \lnot(Q(x, w) \land Q(x, f))\tag{iii}$