Exams and dates

First midterm: Wednesday, February 27 (exam)
Second midterm: Wednesday, April 9 (exam)
Final: Tuesday, May 13 at 12pm (exam)

Lecture notes

I have regrouped the lecture notes into larger files. Otherwise, there is no change; do not reprint them!

01-09 Basic set theory
10-16 Propositional logic
17-27 First order logic

Homework 02, due February 06

Here ≤ denotes "smaller than or equal", and < denotes "strictly smaller than", since I don't know how to type the curly ones.

1. If A is infinite (not in bijection with a finite set), then N ≤ A.
2. Write an explicit bijection between NxN and N.
3. For any A, A<P(A).
4. Think of a category of mathematical objects such that the relevant maps don't satisfy an analog of the Cantor-Schroeder-Bernstein theorem. Prove that you are correct with an example. (If you use a textbook or online resource, you still have to explain your reasoning. Some possibilities are topological spaces, groups, and linear orders.)
5. Prove that there is an injection from A to B iff there is a surjection from B to A. Note explicitly where your proof uses the Axiom of Choice. (Normally in this class, you may use the Axiom of Choice without comment.) The Axiom of Choice states that for any family of nonempty sets {A_i}, there exists a set {a_i} such that for each i, a_i∈A_i.
6. Prove the following are equinumerous:

   • R\N∼R

   • R\Q∼R

   • P(N)∼Sym(N), where Sym(N) denotes the set {f | f:N→N is bijective}

Homework 04, due February 20

1. Suppose that v,v' are truth assignments which agree on every propositional variable appearing in the wff φ. Prove that v(φ)=v'(φ). (Here the bold v means v with a bar over it.) (Argue by induction on the length of the wff φ.)
2. Prove that (A→B) and ((¬B)→(¬A)) are tautologically equivalent.
3. Prove that φ and ψ are tautologically equivalent iff (φ↔ψ) is a tautology.
4. Let η denote a fixed negation of a tautology. Let Σ be a set of wffs. Prove that Σ tautologically implies η iff Σ is unsatisfiable.
5. Let Σ={((¬A)∨B), (B→C), A}. Write down a derivation of C from Σ. (Remember that ∨ can be written using only → and ¬.)
6. Prove the soundness theorem for propositional calculus: If there is a deduction of the wff φ from the set of wffs Σ, then Σ tautologically implies φ.

Homework 05, due March 05

1. Do problems 5,6 from the review for midterm 1.
2. For +4pts (out of 50) on midterm 1, redo ONE of the following two problems. If you more or less got one of them right already, then do the other one!
   (2c) Let C_1/2 denote the set of power series in x with coefficients in Z which converge when x=1/2. Is C_1/2 countable or uncountable?
   (4d) Define α≤β iff (α→β) is a tautology. What are the maximal wffs with respect to this preorder? (Here, since I don't want to rewrite the whole exam problem, let's just define that α is maximal iff α≤β implies β≤α.)

Homework 07, due March 26

Since I can't write the satisfaction symbol, I'm using |= for it.

1. Let L be a language and A be an L-structure with domain A. We say that B⊂A is a definable subset of A iff there exists a wff φ with one free variable x such that

   A|=φ[s] iff s(x)∈B.

   • Prove that {0} and {1} are definable in (N;+).
   • Prove that {2} is definable in (R;+,×).
2. Recall the construction (in class and the notes) of a "non-standard" model A of

   TA={sentences σ : (N;+,×,0,1,<)|=σ}

   Notice that there is a copy of the ordinary natural numbers N contained in A: just identify n∈N with 1+...+1 (n times) in A. This copy of N is called the standard part of A.

   Prove that the standard part of A is an initial segment of A. More precisely, prove that if n∈A is standard and a∈A is nonstandard then n<a. (Hint: first explain why A is a discrete linear order with a least element.)

3. Invent a language L and a set of sentences Σ such that for every natural numberr n, n is even iff there exists an L-structure that satisfies Σ and has size n.

Review for midterm 2

1. Show that + is not definable in (N;×).
2. • Let A be any finite structure and let Σ=Th(A). Prove that Σ does not have any infinite models.

   • Let A be any infinite structure and let Σ=Th(A). Can Σ have any finite models?

3. Prove that wffs φ and ψ are logically equivalent iff the wff (φ↔ψ) is valid.
4. Let L={<,a,b,f}, where < is a binary relation symbol, a,b are constant symbols, and f is a unary function symbol. Let

   Σ={∀x¬(x<x),
   ∀x∀y((x<y)∨(y<x)∨(x=y)),
   ∀x∀y∀z(((x<y)∧(y<z))→(x<z)),
   ∀x∀y((x<y)→∃z((x<z)∧(z<y))),
   ∀x∃y∃z((y<x)∧(x<z)),
   a<b,
   ∀x∀y((x<y)→(fx<fy))}

   • Generally speaking, describe the models of Σ in english.

   • Let σ denote the sentence

   fa<b

   Prove, using counterexamples, that Σ does not logically imply σ and that Σ does not logically imply ¬σ.

5. Show that the class C of 3-colorable graphs with a unique vertex of degree 2 is axiomatizable in the language L={P,Q,R} with three unary predicates.
6. Suppose that A is a structure and that P and Q are definable subsets of A. Let φ(x) be the defining wff for P and let ψ(x) be the defining wff for Q. If φ and ψ are logically equivalent wffs, show that P=Q.
7. Let A be a nonstandard model of arithmetic. Show that every other element is even. (First, say what this means.)
8. Also, study trees.

Homework 10, due April 23

1. Suppose that A is a nonstandard model of TA in which there is a pair p,p+2 of nonstandard prime elements. Show that the twin primes conjecture holds.
2. Prove that first-order axiom groups 3 and 4 are valid, that is, the following wffs are valid.

   • (∀x(α→β))→(∀xα→∀xβ)

   • α→∀xα, where x does not ocurr free in α.

3. Deduce the following wff only from the axioms. At each step, state which axiom group you are using or cite modus ponens. (You should only need the first few axioms, but please re-read them all again carefully from the notes.)

   • (∀x(α→β))→((∀xα)→β)

Review for the final exam

1. • Prove that a countable union of countable sets is countable. (Hint: use the fact that N×N is countable.)

   • Use this to prove that the set Z[x] of polynomials in x with integer coefficients is countable.

2. • Give the definition of a linear ordering.

   • Give the definition of an isomorphism of two linear orderings.

   • Decide whethere the following pairs are isomorphic linear orderings.

   • (Q\Z;<), (Q\{0};<)
   • (Q;<), (Q\(0,1);<)

3. • Give the definition of a tautology in propositional logic.

   • Decide which of the following wffs are tautologies.

   • (Q∨(¬(P→Q)))
   • ((P→(Q→R))↔((P∧Q)→R))

   • Suppose that α is a propositional wff involving only the connectives ∧, ∨, and ¬. Let α' be the wff obtained from α by replacing each occurence of ∧ by ∨, each occurrence of ∨ by ∧, and each occurrence of A by ¬A for A a propositional variable. Prove by induction on the length of α that α and ¬α' are tautologically equivalent.

4. • Use the compactness theorem for first-order logic to derive the completeness theorem for first-order logic.

   • Suppose that Σ is a set of wffs which has an infinite model. Prove that Σ has an uncountable model. (Hint: although we have only established it for countable languages, the compactness theorem is valid for an uncountable language as well.)

5. Let L be the language {<,f}, where < is a binary relation symbol and f is a unary function symbol. Let Σ denote the theory:

   {∀x¬(x<x),
   ∀x∀y((x<y)∨(y<x)∨(x=y)),
   ∀x∀y∀z(((x<y)∧(y<z))→(x<z)),
   ∀x∀y((x<y)→∃z((x<z)∧(z<y))),
   ∀x∃y∃z((y<x)∧(x<z)),
   ∀x∀y((x<y)→(fx<fy))}

   Let σ denote the sentence:

   ∀y∃x(fx=y)

   Prove that there is no deduction from Σ of σ and that there is no deduction from Σ of ¬σ.

6. • Let L be the language {<,P}, where < is a binary relation and P is a unary predicate. Consider the following theory:

   Σ=DLO∪{∀x∀y(x<y→∃z∃w((x<z<y)∧(x<w<y)∧P(z)∧¬P(w))}

   Describe the models of Σ.

   • Using Vaught's test, prove that Σ is a complete theory.

7. A graph is said to be locally finite if every vertex is adjacent to only finitely many other vertices. Prove that the class of locally finite graphs is not axiomatizable.