## Special Homework Assignments (Fall 2012)

1. 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.

2. Let A and B be disjoint sets. Prove that A Bc. [Bc 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.

3. Let d(n) denote the number of diagonals in a convex polygon with n sides (n at least 3). You suspect that d(n) = an2 - 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.

4. 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.

5. 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.

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