Nested Quantifiers Involving Multiple Sets

Question

I'm studying for my midterm and one of the review questions is:

Let S be the set of all students.

Let G be the set of all video games.

Let P(x,y) denote “x has played y”.

Write a nested quantifier to express the sentence “There is a video game that every student has not played

Not looking for the answer... I'm just confused about the different domains and how to express it.

Any help would be greatly appreciated!! Thanks

Answer 1

So, there is a game (so, exists an element of $G$) such that every student (for all elements of $S$) has not played it (it is not true that s played g).

Using this translation, we find the logical expression to be $$\exists g\in G\;\forall s\in S\;\lnot P(s,g)$$