This question is related to discrete mathematics and i need your help to solve it

[Qazi]

Question) Let P(x) be the statement “Student x knows calculus” and let Q(y) be the statement “Class y contains a student who knows calculus.” Express each of these as quantifications of P(x) and Q(y).
a) Some students know calculus.
b) Not every student knows calculus.
c) Every class has a student in it who knows calculus.
d) Every student in every class knows calculus.
e) There is at least one class with no students who know calculus.

 

[stapel]

Question) Let P(x) be the statement “Student x knows calculus” and let Q(y) be the statement “Class y contains a student who knows calculus.” Express each of these as quantifications of P(x) and Q(y).
a) Some students know calculus.
b) Not every student knows calculus.
c) Every class has a student in it who knows calculus.
d) Every student in every class knows calculus.
e) There is at least one class with no students who know calculus.

Please reply with a clear listing of your thoughts and efforts so far, so we can see where you're getting stuck. Thank you!

 

[Qazi]

Please reply with a clear listing of your thoughts and efforts so far, so we can see where you're getting stuck. Thank you!

i have already solved this question by myself but i'm not sure if its right or wrong...so please guide me if there is any mistake, the solved question is given below.


[FONT=&quot]a) Some students know calculus. [/FONT]
[FONT=&quot]∃x P(x) [/FONT]

[FONT=&quot]b) Not every student knows calculus. [/FONT]
[FONT=&quot]∃x ~P(x) [/FONT]
[FONT=&quot][using ~ for "not"] [/FONT]

[FONT=&quot]c) Every class has a student in it who knows calculus. [/FONT]
[FONT=&quot]∀y Q(y) [/FONT]

[FONT=&quot]d) Every student in every class knows calculus. [/FONT]

[FONT=&quot]We haven't actually been given any syntax for "student x is in class y." [/FONT]
[FONT=&quot]We could simply write [/FONT]
[FONT=&quot]∀x P(x) [/FONT]
[FONT=&quot]and argue that this statement, "Every student knows calculus," is equivalent. [/FONT]

[FONT=&quot]Or we could define C(x,y) to represent the statement "Student x is in class y" [/FONT]
[FONT=&quot]and then write [/FONT]
[FONT=&quot]∀y ∀x C(x,y) -> P(x) [/FONT]


[FONT=&quot]e) There is at least one class with no students who know calculus [/FONT]
[FONT=&quot]∃y ~Q(y)[/FONT]

 

[stapel]

i have already solved this question by myself but i'm not sure if its right or wrong...so please guide me if there is any mistake, the solved question is given below.

a) Some students know calculus.
∃x P(x)

"There exists a student...student knows calculus". If this is the syntax you've been told to use, then use it. Otherwise, you might want to include "such that", such as:

. . . . .\(\displaystyle \exists x\, \mbox{ such that }\, P(x)\)

b) Not every student knows calculus.
∃x ~P(x)
(using ~ for "not")

Thank you for defining your notation! Other than an "s.t" for "such that", I'd use the same statement.

c) Every class has a student in it who knows calculus.
∀y Q(y)

I agree.

d) Every student in every class knows calculus.

We haven't actually been given any syntax for "student x is in class y."
We could simply write
∀x P(x)
and argue that this statement, "Every student knows calculus," is equivalent.

Or we could define C(x,y) to represent the statement "Student x is in class y"
and then write
∀y ∀x C(x,y) -> P(x)

Or, if you're not supposed to define any new functionals, maybe, "For all classes y, if student x is in y, then student x knows calculus"? Or maybe the class doesn't matter particularly, so it's something simpler like "For all students x, student x knows calculus"?

e) There is at least one class with no students who know calculus
∃y ~Q(y)

I agree.

 

[Qazi]

thank you so much