Math 3000 Quiz #2

Use logical symbols to rewrite these statements and determine if the argument is valid (that is, the premise [first two statements] implies the conclusion [last statement]).

If everyone in class either studies or is a genius, then someone will pass this course.
If someone passes this course, then they will be allowed to take the next course.
Therefore, if no one is allowed to take the next course, then somebody is not a genius.

Use the predicates S(x) = "x studies", G(x) = "x is a genius", P(x) = "x passes this course" and N(x) = "x is allowed to take the next course", where x is a member of the class C.

Solution:

(xC)(S(x)G(x))(xC) P(x)
(xC) (P(x) N(x))
(xC)~N(x)(xC)(~G(x))
The argument is valid. Notice that
(xC)G(x)(xC)(S(x)G(x))(xC) P(x)(xC) N(x)
So, (xC)G(x)(xC) N(x)
which is the contrapositive of the conclusion.