Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 152 — #160
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Chapter 5
Induction
Problems for Section 5.2
Practice Problems
Problem 5.21.
Some fundamental principles for reasoning about nonnegative integers are:
1. The Induction Principle,
2. The Strong Induction Principle,
3. The Well Ordering Principle.
Identify which, if any, of the above principles is captured by each of the following
inference rules.
(a)
P.0/; 8m: .8k m: P.k// IMPLIES P.m C 1/
8n: P.n/
(b)
(c)
(d)
(e)
P.b/; 8k b: P.k/ IMPLIES P.k C 1/
8k b: P.k/
9n: P.n/
9m: ŒP.m/ AND .8k: P.k/ IMPLIES k m/
P.0/; 8k > 0: P.k/ IMPLIES P.k C 1/
8n: P.n/
8m: .8k < m: P.k// IMPLIES P.m/
8n: P.n/
Problem 5.22.
The Fibonacci numbers F .n/ are described in Section 5.2.2.
Indicate exactly which sentence(s) in the following bogus proof contain logical
errors? Explain.
False Claim. Every Fibonacci number is even.