Every teacher is liked by some student

Question

What is the first order predicate calculus statement equivalent to the following?

"Every teacher is liked by some student"

1. $∀(x)\left[\text{teacher}\left(x\right) → ∃(y) \left[\text{student}\left(y\right) → \text{likes}\left(y,x\right)\right]\right]$
2. $∀(x)\left[\text{teacher}\left(x\right) → ∃(y) \left[\text{student}\left(y\right) ∧ \text{likes}\left(y,x\right)\right]\right]$
3. $∃(y) ∀(x)\left[\text{teacher}\left(x\right) → \left[\text{student}\left(y\right) ∧ \text{likes}\left(y,x\right)\right]\right]$
4. $∀(x)\left[\text{teacher}\left(x\right) ∧ ∃(y) \left[\text{student}\left(y\right) → \text{likes}\left(y,x\right)\right]\right]$

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My attempt :

"Some student likes x" is $ ∃(y) \left[\text{student}\left(y\right) ∧ \text{likes}\left(y,x\right)\right]$

So,

"Every teacher is liked by some student" is

$∀(x)\left[\text{teacher}\left(x\right) → ∃(y) \left[\text{student}\left(y\right) ∧ \text{likes}\left(y,x\right)\right]\right]$

Can you explain in English for each option?

Answer 1

Your answer is correct.

First note that $\text{student}(y)\rightarrow \text{likes}(y,x)$ is equivalent to $ \neg\text{student}(y)\vee \text{likes}(y,x)$.

With that the statements mean the following in plain english:

1. For every teacher there exists a person who is either not a student or likes the teacher.
2. Every teacher is liked by some student.
3. There is some student who likes all the teachers.
4. Everyone is a teacher and for everyone we kind find a person who either likes him or is not a student.