Question

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 1

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"

Answer 2

The expression of the quantifications in English are:

• 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"

Calculations and Parameters:

Based on our prior knowledge of sets, we know that p(x,y) means "Student x has taken class y".

Hence, 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:

f) ∀x∀yP (x, y) simply means  "all the students x took all the classes y"

Read more about domain here:

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