COS 550 F08 HW #7 DUE Thursday 11/06/08


1. (30 points). Partitions and Equivalence Relations.

Please prove the following result.

a. State carefully the definition of an equivalence relation.

b. If R is an equivalence relation on X and a is an element of X, let
   [a] = { b in X | aRb }. Prove that if b is in [a], then a is in [b].

c. If R is an equivalence relation on X, and a and b are in X prove
   that either [a] = [b] or [a] and [b] have an empty intersection.

d. Define a partition of a set.

e. Show that if R is an equivalence relation on X, the collection {
   [a] | a is in X } is a partition of X.

f. Suppose X is a set and P = { S1, S2, ..., Sk } is a partition of X.
   Define a relation R on X by aRb iff there is an i such that a and b
   are both in Si. Prove R is an equivalence relation.

2. (50 points). Fractions.

Let Z be the integers ({0, -1, 1, -2, 2, …}) and Z0 = Z – {0}. Let W =
Z x Z0. Define a relation R on W as follows: (a,b)R(c,d) iff ad = bc.

a.   Prove that R is an equivalence relation.

b.   Define [(a,b)] + [(e,f)] to be [(af+be),bf)]

     1. Prove that + is well-defined on the equivalence classed of R.
        In other words, if you use difference representatives from
        equivalence classes and apply the + operator you end up in the
        same equivalence class.


     2. Prove that + is commutative and associative.

     3. Prove that for every (a,b) there is a (c,d) such
        that [(a,b)] + [(c,d)] = [(0,1)].

c. Define [(a,b)]*[(c,d)] = [(ac,bd)].

     1. Prove that * is well-defined on the equivalence
        classes of R.

     2. Prove that * is commutative and associative.

     3. Prove that for each (a,b) not in [(0,1)], there exists (c,d)
        such that [(a,b)]*[(c,d)] = [(1,1)].

     4. Prove the distributive law:
        [(a,b)]*([(c,d)]+[(e,f)]) = ([(a,b)]*[(c,d)])+([(a,b)]*[(e,f)]).

3. (20 points) Write programs to implement the equivalence relationship 
and operations defined in problem 2 above. Show some non-trivial examples 
that will verify that R is well-defined and that show that the various 
associative, commutative and distributive laws hold.