Note: You are looking at a static copy of the former PineWiki site, used for class notes by James Aspnes from 2003 to 2012. Many mathematical formulas are broken, and there are likely to be other bugs as well. These will most likely not be fixed. You may be able to find more up-to-date versions of some of these notes at http://www.cs.yale.edu/homes/aspnes/#classes.

This assignment is due Friday, September 19th, 2008, at 5:00pm. For due dates of future assignments, see CS202/Assignments.

# 1. Bureaucratic part

This part you will not be graded on, but you should do it anyway.

3. Whether you have ever taken a class that used Grade-o-Matic before.1

4. Anything else you'd like to say.

# 2. Technical part

This part you will be graded on.

## 2.1. Logical equivalences

Which of the following propositions are logically equivalent?

1. p ⇒ ¬r ⇒ ¬q
2. p ⇒ q ⇒ r
3. p ⇒ ¬(q ⇒ r)
4. (p ∧ q) ⇒ r
5. (p ⇒ q) ⇒ r
6. (p ∨ q) ⇒ r

## 2.2. An attempt at realism

Suppose that our universe is the real numbers. Which of the following statements are true? Justify your answers.

1. ∀y∃x y=x+1.
2. ∀y∃x y²>x.

3. ∀y∃x y²<x.

4. ∃y∀x y²<x.

5. ∃x∀y y=x+1.
6. ∃x∀y y²>x.

7. ∃x∀y y²≠x.
8. ∀x∃y y²≠x.

## 2.3. A modeling problem

Consider the following axioms on a unary predicate B and a binary predicate S.

1. ∃x B(x).
2. ∀x∃!y S(x,y).
3. ∀x∀y S(x,y) ⇒ B(x) ⇒ ¬B(y).