CS 390 Test I
February 19, 1998

1(a) Find a string which is not in the language represented by .
[12]
(b) Find a regular expression for the language L defined recursively as follows:

Basis Clause:
Induction Clause: If , then , , and .
Extremal Clause: Nothing is in L unless it is obtained by the above
clauses. [18]

2. Recursively define the set U of all strings of the form . [20]

3. Prove by induction on string y of a's and b's, or by induction on , that . [20]
You may use the following recursive definition of :
Basis Clause:
Inductive Clause: For any string x and a symbol a, .

4. Which of the following statements are true and which are false ? [30]

(a) For an arbitrary string x, , where is the reversal of x.
(b) for any language L.
(c) Suppose A and B are sets, , and are functions.
If f(g(y)) = y for every , then g is onto.
(d) Suppose that A and B are sets and that S and T are subsets of A.
Then .
(e) Let the set W of strings be defined recursively as follows:
Basis Clause:
Inductive Clause: If , then , , , and
.
Extremal Clause: Nothing is in W unless it is obtained by the
Basis and Induction.
Then W is the set of all strings in containing the substring 00.