1. Do exercise 2.3.1 of Winskel at the end of Chapter 2, that is prove
   that the set of partial functions from X to Y forms a domain, with
   the definitions of ordering and lub given on slide 10 (or the
   equivalent definitions given at teh lecture).

2. Which of the following is a domain? (In each case, choose a proper
   definition of "lub"; prove your answer.)

   a. (P(X), \subseteq), where P(X) is the powerset of X, \subseteq is
      the usual subset ordering.

   b. ([0,1], <=), where <= is the usual ordering on the real numbers

   c. ([0,1]\cap Q, <=), \cap is the intersection and Q is the
      rational numbers

   d. (Sigma^*, c=), where Sigma^* is the set of words over the
      alphabet Sigma := {a,b} and c= is the prefix ordering, defined
      by w c= wv for all w,v in Sigma^*

3. Do exercise 2.3.2 of Winskel at the end of Chapter 2, that is prove
   that the function f_b,c in the definition of the denotational
   semantics of while B do C is continuous.

4. Let (d_i)_{i>=0} and (e_i)_{i>=0} be chains in a domain D. Suppose
   that (d_i)_{i>=0} "is majorized by" (e_i)_{i>=0}, that is:
     forall i, exists j (d_i <= e_j)
   Prove that lub_{i>=0} d_i  <=  lub_{i>=0} e_i 

5. Suppose that in the domain D, "all chains are eventually constant",
   that is, if d_0 <= d_1 <= d_2 <= is a chain then
       d_n = d_{n+1} = d_{n+2} = ... for some n.
   Show that every monotone f : D -> D is continuous