Some remarks concerning and, implies and quantifiers in problem 10 in section 3.1.

10a Let c(x) be x is in this class and r(x) be x owns a red convertible Linda, a student in this class, owns a red convertible “.should be translated as

c(Linda) and r(Linda). This statement indicates that following facts are true. First that Linda is in this class, and second that Linda owns a red convertible. Translating it as c(Linda) implies r(Linda) is incorrect, since this says only that if Linda is in the class then she owns a red convertible. It does not imply that Linda is actually in the class, which is what we want to assert.

10b) I messed up the solution to this problem. Let me try to sort things out. Let r(x) be x is one of the five roommates, d(x) be x has taken a course in discrete mathematics Each of the five roommates has taken a course in discrete mathematics should be translated For any x (r(x) implies d(x)). This says that anyone who is one of the five roommates has taken a course in discrete mathematics. It cannot be translated as

for any x( r(x) and d(x)) which would say everyone is one of the five roommates and everyone has taken a course in discrete mathematics.

Hypotheses and conclusions are statements asserted or proved to be true. To have a truth value, they must either be instantiated, like c(Linda) and r(Linda), or quantified like

for all x (r(x) implies d(x)). Without the quantifiers, r(x) implies d(x) has no truth value and cannot be a hypothesis.