Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 223 — #231
7.5. Induction in Computer Science 223
Fact.
.rs D uv AND jrj D juj/ IFF .r D u AND s D v/ (7.28)
r .s t/ D .r s/ t (7.29)
rev.st/ D rev.t/ rev.s/ (7.30)
(a) Prove that s D rev.s/ for all s 2 RecPal.
(b) Prove conversely that if s D rev.s/, then s 2 RecPal.
Hint: By induction on n D jsj.
Problem 7.8.
Let m; n be integers, not both zero. Define a set of integers, L m;n , recursively as
follows:
Base cases: m; n 2 L m;n .
Constructor cases: If j; k 2 L m;n , then
1. j 2 L m;n ,
2. j C k 2 L m;n .
Let L be an abbreviation for L m;n in the rest of this problem.
(a) Prove by structural induction that every common divisor of m and n also divides
every member of L.
(b) Prove that any integer multiple of an element of L is also in L.
(c) Show that if j; k 2 L and k ¤ 0, then rem.j; k/ 2 L.
(d) Show that there is a positive integer g 2 L that divides every member of L.
Hint: The least positive integer in L.
(e) Conclude that g from part (d) is gcd.m; n/, the greatest common divisor, of m
and n.
Problem 7.9.