Show all work for credit. Do all your work neatly. I will not grade work I can not read. Write your name on each sheet you turn in. I will not grade any work done on the test sheet. Please staple all your work and the test together and turn it in Wednesday July 26th at 9:20 am. Good Luck.

1. Let S = N (the set of natural numbers) and let R be the relation on S defined by (m,n) Î R if m² - n² is even. Show that R is an equivalence relation and describe the resulting equivalence classes.

2. A set of four coins is selected from a box containing 5 dimes and 7 quarters.

   1. Find the number of sets of four coins.
   2. Find the number of sets in which two are dimes and two are quarters.

3. Prove that two graphs are not isomorphic if one is connected and the other is not.

4. If all the vertices of a planar connected graph have degree 4 and the number of edges is 12, into how many regions does it divide the plane?

5. Prove that every graph with exactly one odd cycle has a proper coloring with 3 colors.

6. Find a (7,7,4,4,2) design.

7. Prove that a balanced incomplete uniform design is regular.

8. Does there exist a BIBD with parameters b = 20, v = 18, k = 9, and r = 10?

9. Draw all non-isomorphic graphs with four vertices.

10. Use the binomial theorem to prove that 

                  .

11. Prove that 1 ·1! + 2 ·2! + 3 ·3! + ... + n·n! =  (n + 1)! - 1.