Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 219 — #227
7.5. Induction in Computer Science 219
Problems for Section 7.1
Practice Problems
Problem 7.1.
The set OBT of Ordered Binary Trees is defined recursively as follows:
Base case: hleafi is an OBT, and
Constructor case: if R and S are OBT’s, then hnode; R; Si is an OBT.
If T is an OBT, let n T be the number of node labels in T and l T be the number
of leaf labels in T .
Prove by structural induction that for all T 2 OBT,
Class Problems
l T D n T C 1: (7.24)
Problem 7.2.
Prove by structural induction on the recursive definition(7.1.1) of A that concatenation
is associative:
.r s/ t D r .s t/ (7.25)
for all strings r; s; t 2 A .
Problem 7.3.
The reversal of a string is the string written backwards, for example, rev.abcde/ D
edcba.
(a) Give a simple recursive definition of rev.s/ based on the recursive definitions
7.1.1 of s 2 A and of the concatenation operation 7.1.3.
(b) Prove that
rev.s t/ D rev.t/ rev.s/; (7.26)
for all strings s; t 2 A . You may assume that concatenation is associative:
for all strings r; s; t 2 A (Problem 7.2).
.r s/ t D r .s t/