Page 319 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Closures Again 305
(b) Prove that S \ i A ={(a, b) ∈ A × A | there is a simple R-path from
a to b}.
12. Suppose R is a relation on A. In this problem we find a relationship
between distance, as defined in exercise 9, and the lengths of R-paths,
which were discussed in exercises 10 and 11.
(a) Suppose d(a, b) = n and a = b. Prove that there is a simple R-path
from a to b of length n.
(b) Suppose d(a, a) = n. Prove that there is an R-path f from a to a of
length n such that ∀i < n∀ j < n( f (i) = f ( j) → i = j). (In other
words, f is simple, except for the fact that f (0) = f (n) = a.)
13. Suppose R is a relation on A, S is the transitive closure of R, and A has
m elements. Prove that
2
m
n
S = R ∪ R ∪ ... ∪ R =∪{R | 1 ≤ n ≤ m}.
(Hint: Use exercise 12. What is the maximum possible length of a simple
R-path?)
∗
14. There is another proof in the introduction that could be written more rig-
orously using induction. Recall that in the proof of Theorem 4 in the intro-
duction we used the fact that if n is a positive integer, x = (n + 1)! + 2,
and 0 ≤ i ≤ n − 1, then (i + 2) | (x + i). Use induction to prove this. (We
used this fact to show that x + i is not prime.)