The first three problems come from page 39 of the Gries & Schneider text.
1. Do Exercise 2.6.
2. Do parts (a) through (i) of Exercise 2.7.
The "primitive components" of the English sentences are to be named as indicated below.
xly : x < y xey : x = y xgy : x > y
ylz : y < z ygz : y > z
vew : v = w
(Note that x ≥ y (mentioned in parts (h) and (i)) need not be named, since it can be expressed as xey ∨ xgy .)
3. Do Exercise 2.9 (The Tardy Bus Problem).
4. In each step in the proof below, Leibniz (with an assist from Substitution) is applied. That is, in going from each line to the next, some subexpression E is replaced by an expression F such that E = F (or possibly F = E) is an instantiation of (i.e., the conclusion of applying Substitution to) one of the theorems in our arsenal. (That is, E = F (or else F = E) is P[r:=Q], where P is a theorem in our arsenal, r is a list of variables, and Q is a list (of equal length) of expressions.)
Here, the only theorems used are axioms (3.1), (3.2), and (3.3). Fill in each "hint" so as to inform the reader which theorem (and which instantiation of it) was used. The first hint is filled in for you. Also, somehow highlight the subexpressions E and F in each step. In the first step below, the former is in boldface and the latter is in italics.
(p = q) = (true = (q = r))
= < (3.3), with q:=p >
(p = q) = ((p = p) = (q = r))
= < >
(p = q) = (p = (p = (q = r)))
= < >
(p = q) = (p = ((p = q) = r))
= < >
(p = q) = (((p = q) = r) = p)
= < >
(p = q) = ((p = q) = (r = p))
= < >
((p = q) = (p = q)) = (r = p)
= < >
true = (r = p)
= < >
r = p
The remaining problems come from page 62 (at the end of Chapter 3) of the Gries & Schneider text. In proving a theorem, you may make use of only lower-numbered theorems (or "metatheorems").
5. Do Exercise 3.7, which asks you to prove (3.11) ¬p = q = p = ¬q in three different ways:
(a)
transform ¬p = q into
p = ¬q
(b) transform ¬p = p into q = ¬q
(Note: For this to constitute a proof of (3.11), we need to show also
that (3.11) = ((¬p = p) = (q = ¬q)).
But, quite obviously, this equivalence holds by the symmetric property
of "=". Do the formal proof on scrap paper to amuse yourself!)
(c) transform ¬p into q = p = ¬q
For this homework, you are required to do only two of these three.
6. Do Exercise 3.12, which asks you to prove the associativity of ≠:
using the heuristic of Definition Elimination (3.23). (Note that ≠ is used here rather than ≡ with a line through it simply because there is no such "character entity reference" in HTML.)