0

I know the question might be repeated but I am not able to understand this particular question regarding nested quantifier. So I need some clarification and guidance on this.

If: $$A(x,y) = \text{x attends y's office hour} \\ \text{where the domain of discourse is all students } \times \text{all teachers}$$

For this sentence: "All teachers have at least one student attend their office hours"

Is $$\forall y \exists x A(x,y) \equiv \exists x \forall y A(x,y)$$

The left part is the right answer from the book, but I don't know why the one on the right doesn't work. And what is the difference?

Thanks!

Yan Zhuang
  • 569
  • 1
  • 5
  • 20
  • Presumably the domain is actually Students $\times$ Teachers, since $x$ is always a Student and $y$ is always a Teacher. Accordingly, your statement (the right hand) would translate as "there is some student who attends the office hours of every teacher". – lulu Sep 19 '20 at 18:49
  • So when we have a nested quantifier we cannot change the order of the quantifier? Or it depends? – Yan Zhuang Sep 19 '20 at 18:53
  • 1
    Indeed, you can not freely change the order of quantifiers. – lulu Sep 19 '20 at 18:56
  • Thanks a lot !! – Yan Zhuang Sep 19 '20 at 18:58
  • 1
    See this nice [1-page document by S. Marc Cohen](https://faculty.washington.edu/smcohen/320/QuantifierOrder.pdf). Also, [Intuitive Reason that Quantifier Order Matters](https://math.stackexchange.com/q/491783/13130). – Dave L. Renfro Sep 19 '20 at 19:01
  • These are really good notes. Thanks a lot! – Yan Zhuang Sep 19 '20 at 19:05

0 Answers0