1. (a) Express the following sentence in a formal language:
If every player beats some opponent in every tournament that features some champion from every country holding some title, then some player beats every opponent in some tournament that features every champion from some country holding every title.
(b) Draw a formula tree for the preceding sentence.

2. Draw a formula tree for the sentence that expresses Goldbach's Conjecture. (See one of the handouts from August 26 for the statement of Goldbach's Conjecture.)

3. Draw a formula tree for the sentence that expresses that $f$ is a continuous function. (See one of the handouts from August 26 for the formal statement expressing that $f$ is continuous.)

We will work on Problem 1 in class. You should submit Problems 2 and 3 by the due date.

HW1 solutions.pdf
HW1 solutions.tex

1. Exercise 10 of Section 1.5.

2. Exercise 2 of Section 1.5.

3. Exercise 5 of Section 1.5.

We will work on Problem 1 in class. You should submit Problems 2 and 3 by the due date.

HW2 solutions.pdf

1. Exercise 1 of Section 1.7.

2. Exercise 3 of Section 1.7.

3. Show that $\Gamma\models (\alpha\to \beta)$ holds if and only if $\Gamma\cup \{\alpha\}\models \beta$ holds. (Hint: Assume that one of them fails and that the failure is witnessed by a valuation $v$. Then show that $v$ witnesses that the other fails.)

We will work on Problem 1 in class. You should submit Problems 2 and 3 by the due date. HW3 solutions.pdf

Read Section 2.1.

1. Exercises 1(e) and 1(f) of Section 2.1.

2. Exercises 1(b) and 1(d) of Section 2.1.

3. Exercise 6 of Section 2.1.

We will work on Problem 1 in class. You should submit Problems 2 and 3 by the due date.

HW4 solutions.pdf

Read pages 80-89 of Section 2.2.

1. Complete this handout.

2. Exercise 1 of Section 2.2.

3. Exercise 2 of Section 2.2.

We will work on Problem 1 in class. You should submit Problems 2 and 3 by the due date.

HW5 solutions.pdf

Check the due date!
Read pages 90-100 of Section 2.2.

1. Exercise 12(c) of Section 2.2.

2. Exercise 15 of Section 2.2.

3. Exercise 28(a) of Section 2.2.

We will work on Problem 1 in class. You should submit Problems 2 and 3 by the due date.

HW6 solutions.pdf

Check the due date!
Read pages 90-100 of Section 2.2.

1. Exercise 18(a) of Section 2.2, first sentence only.

2. Exercise 27 of Section 2.2.

3. Give an example of a universal sentence and an embedding, where the embedding does not preserve satisfaction of the universal sentence. Then, give an example of an existential sentence and an embedding, where the embedding does not reflect satisfaction of the existential sentence.

We will work on Problem 1 in class. You should submit Problems 2 and 3 by the due date.

HW7 solutions.pdf

Check the due date!
Read pages 109-128 of Section 2.4.

1. Exercise 4 of Section 2.4.

2. Exercise 6(a) of Section 2.4.

3. Exercise 6(b) of Section 2.4.

We will work on Problem 1 in class. You should submit Problems 2 and 3 by the due date.