CS 390 Test I Solutions
February 19, 1998

1(a) For example 001, 1101.
(b) For example tex2html_wrap_inline55

2. Basis Clause: tex2html_wrap_inline57
Inductive Clause: If tex2html_wrap_inline59 then tex2html_wrap_inline61 and tex2html_wrap_inline63 .
Extremal Clause: Nothing is in U unless it is obtained from the above two clauses.

3. Proof by induction on y.
First let us note that tex2html_wrap_inline69 can be defined recursively as follows:
Basis Clause: tex2html_wrap_inline71 .
Inductive Clause: For arbitrary string x of tex2html_wrap_inline69 , xa and xb are in tex2html_wrap_inline69 .
Extremal Clause: Nothing is in the set unless it is obtained from the above two clauses.

The proof of the equality in question is going to be proven for an arbitrary fixed x by induction on y. Thus the statement to be proven is for an arbitrary fixed string x, and an arbitrary string y, tex2html_wrap_inline93 holds.
The proof mirrors the recursive definition of tex2html_wrap_inline69 .

Basis Step: tex2html_wrap_inline97 .
Inductive Step: Assume that for an arbitrary string y, tex2html_wrap_inline93 holds.
--- Induction Hypothesis

Then for an arbitrary symbol a, tex2html_wrap_inline105 .
But by induction hypothesis tex2html_wrap_inline107 . Since tex2html_wrap_inline109 , tex2html_wrap_inline111 holds, which is what we needed.

4. False for (c) and (d). The rest are all true.