Q1: Simplification of Formulas
There are many ways to represent a formula. For example: (p \/ (p => q))
is equivalent to true. For each of the following, try to find the simplest
equivalent formula. By simplest, we mean the one with the least number of
connectives and parentheses. You can use any unary or binary connective
we introduced in class.
  1. ((p => q) \/ r) /\ ~q
  2. (p \/ r) /\ (q => r) /\ (r => p)
  3. (~q => ~p) /\ (~p => q)


Q2: Word Problems
The problems below consist of some assumptions followed by a conclusion.
Formalize and analyze the statements using propositional logic. Does the
conclusion follow from the assumptions? Clearly explain your solution.
For example, suppose you were asked to formalize.
  
  1. Jack will ask Anne to the dance only if Nick asks Fiona. Jack did not
     ask Anne to the dance. Therefore Nick did not ask Fiona.
  
  2. If Jeremy fails the midterm, he will get a C in the class. If he gets a C
     in the class, he won't graduate cum laude. Jeremy fails the midterm.
     Therefore he will not graduate cum laude.
  
  3. Dan has eggs for brunch only if he eats either bacon or sausage. Dan
     eats both bacon and sausage. Therefore he eats eggs.