Math 570: Mathematical Logic
Fall Semester 2004
Prof. Ward Henson
Friday, September 3, 2004
Problem Set 1 – Due in class Monday, September 13, 2004.
There are five problems (25 points each) and you should do four of them.
To earn full credit requires a careful writeup of each problem, taking care to
justify everything you claim and to explain your ideas clearly. In Problems 1,2,3
you may NOT use the Completeness Theorem, but you may use the Deduction
Lemma as well as Lemma 2.2.1 and Corollary 2.2.7.
1.1. Problem. Let Abe a set of atoms and suppose Σ ⊆Prop(A) is a maxi-
mal consistent set of propositions. Let t:A→ {0,1}be the unique truth assign-
ment satisfying t(a) = 1 ⇔a∈Σ for all a∈A. Write out a complete and careful
proof that for any p∈Prop(A), one has t(p)=1⇔p∈Σ. (Some of this was
done in class, but you should include that in your writeup. The basic structure
of your argument can be similar to the proof of Lemma 2.2.10, although the
statements of the results to be proved are technically different. Note that even
in the text, Case 3 of that proof was left to the reader.)
1.2. Problem. Let Abe a set of atoms. To each symbol sassign a weight,
denoted w(s), as follows: if sis an atom or >or ⊥, then w(s) = −1; w(¬) = 0;
w(∧) = w(∨) = 1.
Give a proof by induction on propositions of the following statement: If s1, . . . , sm
are symbols such that the word s1. . . sm∈Prop(A), then w(s1) + · · · +w(sm) =
−1.
1.3. Problem. Let Abe a set of atoms, Σ ⊆Prop(A), and p, q ∈Prop(A).
Prove the following statements:
(a) {p, q}`∧pq;
(b) if Σ ∪ {p} ` qand Σ ∪ {q} ` p, then Σ `(p↔q);
(c) if Σ ∪ {p} ` q, then Σ ∪ {¬q}`¬p;
(d) ¬ ∨ pq ` ∧¬p¬q.
1.4. Problem. Let Abe a finite set of atoms, with nelements.
(a) For any set Sof truth assignments (from Ato {0,1}), show that there is
a proposition p∈Prop(A) such that Sis exactly the set of truth assignments
t:A→ {0,1}such that t(p) = 1.
(b) A subset {p1, . . . , pk}of Prop(A) is independent if {pi|i∈F} 6|=pjholds
whenever 1 ≤j≤k,F⊆ {1, . . . , k}, and j6∈ F. Determine the maximum size
of an independent subset of Prop(A).
1.5. Problem. Let Abe a set of atoms and let Σ and ∆ be subsets of Prop(A).
Assume that there does not exist any truth assignment t:A→ {0,1}which is
simultaneously a model of Σ and a model of ∆. Show that there exists p∈
Prop(A) such that Σ `pand ∆ ` ¬p.