Answer: Hello mate!
we know that p(x,y) means "Student x has taken class y"
and the used symbols are:
∃: this means "existence", you use this symbol to say that there exists at least one object that makes true the sentence.
∀: this means "for all", you use this symbol to say that the sentence is true for all the elements, then:
a) ∃x∃yP (x, y)
"exist at least one student x, that took at least one class y"
b) ∃x∀yP (x, y)
"exist at least one student x, that took all the classes y"
c) ∀x∃yP (x, y)
"every student x, took at least one class y"
d) ∃y∀xP (x, y)
"exist at least one class y, that has been taken by all the students x"
e) ∀y∃xP (x, y)
"for every class y, there is at least one student x that took the class"
f) ∀x∀yP (x, y)
"all the students x took all the classes y"
