Assignment 1

CS 270/680 Mathematical Foundations of Computer Science
Instructor: Jeremy Johnson 
Due date: (Beginning of class on Oct. 5)

  1. Exercise 2.3.2 from the text
  2. Exercise 2.3.7 from the text
  3. Exercise 2.3.8 from the text
  4. Exercise 2.6.7 from the text
  5. Exercise 2.6.8 from the text

Graduate Student Problem

  1. Exercise 2.3.13 from the text

Bonus Problem 

In a 3-dimensional hypercube (ordinary cube), there are 8 vertices (0-dimensonal cubes), 12 edges (1-dimensional cubes), and 6 faces (2-dimensional cubes).  Euler's formula states that V - E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces.  In this question, you will generalize this result to an n-dimensional hypercube.

In class, we showed that the number of edges E(n) (1-dimensional hypercubes) in an n-dimensional hypercube satisfies the recurrence relation: E(n) = 2*E(n-1) + 2^(n-1), for n > 1, and E(1) = 1.  The solution to this recurrence is n*2^(n-1).  Similarly,  the number of vertices V(n) satisfies the recurrence V(n) = 2*V(n-1), for n > 0, and, V(0) = 1, and that V(n) = 2^n.
  1. Derive a recurrence relation for, C(n,k), the number of k-dimensional cubes in a n-dimensional cube.  Solve this recurrence to obtain a closed-form solution as a function of n and k.  Hint, you will need to use the binomial coefficient binom(n,k) = n!/(k!*(n-k)!), where n!  n*(n-1)*...*2*1.
  2. Find and prove a relationship between the functions C(n,k) for k=0,...,n that generalizes Euler's formula.  Hint you should review the binomial theorem.

How to submit

Students should submit their solution electronically.  Use a word processor to create your solution and submit either 1) a text file, 2) a word document, or 3) a pdf file.  You should submit your assignment using WebCT.  Bring a paper copy of your homework to the class meeting when it is due.  WebCT will be set up to accept assignments on Thursday.