Let P (x, y) be the statement "Student x has taken class y," where the domain for x consists of all students in your class and for y consists of all computer science courses at your school. Express each of these quantifications in English.

a) ∃x∃yP (x, y)

b) ∃x∀yP (x, y)

c) ∀x∃yP (x, y)

d) ∃y∀xP (x, y)

e) ∀y∃xP (x, y)

f) ∀x∀yP (x, y)

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"