# Homework Assignment #4

#### Due on paper in lecture, Friday 9 March 2012.

There are eight questions for 60 total points plus 10 extra credit. All but two are from the textbook, Mathematical Foundation for Computer Science. Note that the book has both Exercises and Problems -- make sure you are doing a Problem and not the Exercise with the same number. The number in parentheses following each problem is its individual point value.

Correction in purple (to Problem D-1) added 5 March.

Correction in green (to Problem 3.6.2) added 6 March.

Correction in orange (again to D-1) added 9 March.

• Problem 3.5.1 (5)

• Problem 3.6.2 (5) Instead of "any number" it should say "any positive natural". You can't write zero or negative integers as products of prime powers.

• Problem 4.1.3 (10)

• Problem 4.3.1 (10)

• Problem D-1 (10): Let S(n) be the sum, for i from 1 through n, of (-1)ii2. For example, S(4) = -1 + 4 - 9 + 16 = 10. Find a polynomial whose value is S(n) and prove by induction that your result is correct. (Your polynomial may include a (-1)n, for example.)

• Problem 4.4.4 (10)

• Problem 4.4.5 (10)

• Problem D-2 (10XC) Prove by induction that the number of possible Hasse diagrams for the relation S in Problem 2.10.5 (d) is 2n-2, where n is the number of teams in the tournament.