The exercises referred to by number come from pages 22-23 of the Gries & Schneider textbook.
1. Do Exercise 1.7, parts (b) through (e). Each answer should show an instantiation of the Leibniz inference rule, together with a statement telling to what each of X, Y, E, and z was instantiated in order to arrive at it. For example, the answer to part (a) is
4*x+y = 4*(x+2)+y |
at which one arrives by the instantiation X:x, Y:x+2, E:4*z+y, z:z. (Note that none of the given instantiations of Leibniz mentions the variable z; hence you can always provide a solution in which z is instantiated to itself.)
2. Do Exercise 1.8, parts (b) through (e).
The solution to (a) is b+c+y+w.
(Note: Depending upon which printing of the book you have, you may or may
not have to swap the two sides of X=Y in parts (b) and (e).
In the image to which there is a hyperlink above, you DO have to do a
swap.)
3. Do Exercise 1.9, parts (b) through (e). The solution to (a) is E:(x+y)*z, X:x+y, Y:y+x.
4. Let f : bool × bool → bool be a two-argument boolean function. We say that f is reflexive if f(p,p) is true for both possible values of p (i.e., true and false). We say that f is symmetric if f(p,q) = f(q,p) for all possible combinations of values of p and q (of which there are four, of course).
Exactly two among the sixteen two-argument boolean functions are reflexive but not symmetric. Identify them.
5. Formalize the following argument. That is, identify the atomic propositions and give each one a unique name (i.e., in the form of a boolean variable). Use those names, together with boolean operators, to express each premise, as well as the conclusion. You are not expected to indicate whether or not the argument is valid.
Exactly one among Barb and Carl is dead. If Barb is alive, then either Dave is retired or Edith plays golf. If Dave is retired, then at most one among Fred and Arnie lives in New York. If Arnie lives in New York and Edith does not play golf, then Fred does not live in New York. Edith does not play golf! It follows that Carl is alive.