# Exercise

## Nested Quantifiers

##### Problem A

Let $C(x,y)$ mean that $x$ is enrolled in class $y$. Express these into **simple** English sentences:

- $\forall x \exists y C(x,y)$
- $\forall y \exists x C(x,y)$
- $\exists x \forall y C(x,y)$
- $\exists y \forall x C(x,y)$
- $\forall y \forall x C(x,y)$
- $\exists y \exists x C(x,y)$

##### Problem B

Let:

- $W(x,y)$ means that student $x$ has visited website $y$.
- The domain for $x$ is everyone in the world
- The domain for $y$ is all websites

Express these into **simple** English statements (the last 3 are kinda challenging, take your time)

- $W("\text{Jimbob Ghostkeeper}", "\text{heeeeeeeey.com}")$
- $\exists x W(x,"\text{thatsthefinger.com}")$
- $\exists x W(\text{"Kobe Buffalomeat"},x)$
- $\exists x (W(\text{"Pope McCorkle III"},x) \land \text{"Shamus Beaglehole"},x))$
- $\exists x \forall y (x \neq \text{"Nohjay Nimpson"} \land W(\text{"Nohjay Nimpson"},y) \to W(x,y))$
- $\exists x \exists y \forall z((x \neq y) \land W(x,z) \land W(y,z))$
- $\exists x \exists y \forall z((x \neq y) \land W(x,z) \leftrightarrow W(y,z))$

##### Problem C

Let:

- $S(x)$ mean $x$ is a student
- $F(x)$ mean $x$ is a faculty member
- $A(x,y)$ mean $x$ has asked $y$ a question
- The domain for all variables are all people in the school

Express these into quantified statements

- Leo Moses Spornstarr has asked Prof. Barkevious Mingo a question
- Every student has asked Prof. Courage Shabalala a question
- Some students have not asked any of the professors a question
- There is a faculty member who has asked every other faculty member a question
- All students have been asked by some faculty members
- There is a student who has asked all faculty members a question
- No student have not been asked questions by Prof. Andy Brandy Casagrande