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)

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"

2. angles qrt and qst both measures 156 degrees.

step-by-step explanation:  

we have been given a diagram of a circle and we are told that measure of arc qvt is 156 degrees.  

we can see from our given diagram that angle qrt and angle qst subtends to arc qvt.

since we know that angles subtended by the same arc at the circumference are always equal.

by angle subtended by same arc theorem, measure of angle qrt and angle qst will be equal to the measure of arc qvt.

therefore, angles qrt and qst both measures 156 degrees and 2nd statement is the correct choice.

60°

step-by-step explanation:

the measure angle of c in an isosceles triangle abc is 60° because in isosceles triangle,the measures of all the angles is 60°.