Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 225 — #233
7.5. Induction in Computer Science 225
hblacki 2 RBT.
Constructor cases: A; B are RBT’s, then
if A; B start with black, then hred; A; Bi is an RBT.
if A; B start with red, then hblack; A; Bi is an RBT.
For any RBT T , let
r T be the number of red labels in T ,
b T be the number of black labels in T , and
n T WWD r T C b T be the total number of labels in T .
Prove that
If T starts with a red label; then n T
3 r T 2n T C 1
; (7.31)
3
Hint:
n=3 r IFF .2=3/n n r
Exam Problems
Problem 7.12.
The Arithmetic Trig Functions (Atrig’s) are the set of functions of one real variable
defined recursively as follows:
Base cases:
The identity function id.x/ WWD x is an Atrig,
any constant function is an Atrig,
the sine function is an Atrig,
Constructor cases:
If f; g are Atrig’s, then so are
1. f C g
2. f g