Problem Set 2: 

Course:		CS 407 -- Codes and Internet Security
Posting Date:   September 21, 1998
Due Date:       September 28, 1998
Builds On:

   For an explanation of most of the pseudo-TeX used here, please consult 
problem set 1.  The new pseudo-Tex symbols are \emptyset for the empty set and 
\neq for not = (C != or PASCAL <>).

   There are three problems in this problem set.  The words "show", "prove" and 
"demonstrate" all have the same meaning.

1. Let I be the unit interval, I = (0, 1) = {t : 0 <= t <= 1}.  Let X = I x I = 
{(x, y) : x \in I and y \in I}.  Let \cal P = {X_t : t \in I}, where X_t = {(t, 
y) : y \in I}.  Show that the X_t are pairwise disjoint, that is show that X_s 
\intersect X_t = \emptyset for all s, t \in I with s \neq t.

2. Prove that if \cal P is a partition of a set X, then the set X must itself 
be non-empty.  (This is a one-liner).

3. Let \cal P = {X_i : i \in I} be a partition of X.  Prove that each point of 
X belongs to X_i for one and only one value of i.