11.3 Prove that each argument is valid by replacing each proposition with a variable to obtain the form of the argument. Then use the rules of inference to prove that the form is valid. (b) If it was not foggy or it didn't rain (or both), then the race was held and there was a trophy ceremony. The trophy ceremony was not held. ------------------------------------ It rained.

(c) If I work out hard, then I am sore. If I am sore, I take an aspirin. I did not take an aspirin. ----------------------------------- I did not work out hard.

Please show steps

7.6

In the following question, the domain of discourse is a set of male patients in a clinical study. Define the following predicates:

P(x): x was given the placebo D(x): x was given the medication A(x): x had fainting spells M(x): x had migraines

Suppose that there are five patients who participated in the study. The table below shows the names of the patients and the truth value for each patient and each predicate: P(x) D(x) A(x) M(x) Frodo T F F T Gandalf F T F F Gimli F T T F Aragorn T F T T Bilbo T T F F

For each of the following quantified statements, indicate whether the statement is a proposition. If the statement is a proposition, give its truth value and translate the expression into English.

(c) x M(x) D(x)

(d) x (A(x) M(x))