Here are the /Solutions.

This assignment is due Tuesday, September 13th, 2005 at 11:00pm. 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.

Send me email. My address is `<aspnes@cs.yale.edu>`. In your message, include:

- Your name.
- Your status: whether you are an undergraduate, grad student, auditor, etc.
**Which course you are taking.**Whether you have ever taken a class that used Grade-o-Matic before.

^{1}- Anything else you'd like to say.

# 2. Technical part

This part you will be graded on.

Prove or disprove: (p \/ q) => (q \/ r) is logically equivalent to (p \/ ¬q) => r.

Prove or disprove: for all natural numbers n, n

^{2}+2n is a multiple of 3.Prove or disprove: for all natural numbers n, n

^{3}+2n is a multiple of 3.Prove or disprove: for all sets A and B, ∅∈A => ∅⊆B.

Prove by induction on n that, for any real number x ≥ -1 and any integer n ≥ 1, (1+x)

^{n}≥ 1 + nx. (Hint: use the fact that a ≥ 0 and b ≥ c implies ab ≥ ac, where a, b, and c are all real numbers.)

Clarification added 2005-09-06: for problems 2 and 3, you should assume that 0 is a natural number (and that it is a multiple of 3).

Ulterior motive: The information in your email will be used to create an account for you in Grade-o-Matic. (1)