- Let there be 9 points in 3-space with integer coefficients.
Prove that there is a pair of these points whose line segment
contains an interior point whose coordinates are integers.
You are to fill in all the missing details of the proof outline given in class.

First attempt due by

**10/2**. Assignment to be finished by**10/23**. - Let A and B be disjoint sets. Prove that A B
^{c}. [B^{c}is the complement of B]. Write your proof**without**using any symbols except for the names of sets and elements (*the assignment will be returned if any other symbols appear*).First attempt due by

**10/9**. Assignment to be finished by**10/23**. - Let d(n) denote the number of diagonals in a convex polygon with n sides (n at least 3). You
suspect that d(n) = an
^{2}- bn for some constants a and b. Determine these constants by examining some small cases (n = 3,4,5, ...). Then use induction to prove that your formula is correct.First attempt due by

**10/23**. Assignment to be finished by**11/6**. - Define a relation R on the set × of ordered pairs of real numbers by:

(x,y) R (u,v) iff 3x - y = 3u - v.

Prove that R is an equivalence relation and describe the equivalence class containing (4,5). Give a geometric description of the equivalence class containing (a,b).First attempt due

**11/13**. Assignment to be finished by**11/29**. - Prove that every set with a denumerable subset has the property that it is equivalent to a
proper subset of itself.
Note: This proper subset need not be the denumerable subset.

Hint: Start by proving that a denumerable set has the property.

First attempt due

**11/27**to be finished by**12/11**.