Page 318 - HOW TO PROVE IT: A Structured Approach, Second Edition

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304 Mathematical Induction
6. Suppose F is a family of functions from A × A to A, and B ⊆ A. Com-
bining the ideas in exercises 3 and 4, we say that the closure of B under F
is the smallest set C such that B ⊆ C ⊆ A and for all f ∈ F, C is closed
under f , if there is such a smallest set.
(a) Prove that the closure of B under F exists.
(b) Let f : R × R → R and g : R × R → R be defined by the formulas
f (x, y) = x + y and g(x, y) = xy. Prove that the closure of Q ∪
√ √
{ 2} under { f, g} is the set {a + b 2 | a ∈ Q, b ∈ Q}. (This set is
√ √
called Q with 2 adjoined, and is denoted Q( 2).)
√
3
(c) With f and g defined as in part (b), what is the closure of Q ∪{ 2}
under { f, g}?
7. Suppose R and S are relations on A and R ⊆ S. Prove that for every
n
n
positive integer n, R ⊆ S .
8. Suppose R and S are relations on A and n is a positive integer.
∗
n
n
n
(a) What is the relationship between R ∩ S and (R ∩ S) ? Justify your
conclusions with proofs or counterexamples.
n
n
n
(b) What is the relationship between R ∪ S and (R ∪ S) ? Justify your
conclusions with proofs or counterexamples.
9. Suppose R is a relation on A and S is the transitive closure of R.
If (a, b) ∈ S, then by Theorem 6.5.2 there is some positive integer
n
n such that (a, b) ∈ R , and therefore by the well-ordering principle
(Theorem 6.4.4), there must be a smallest such n. We define the distance
n
from a to b to be the smallest positive integer n such that (a, b) ∈ R , and
we write d(a, b) to denote this distance.
(a) Suppose that (a, b) ∈ S and (b, c) ∈ S (and therefore (a, c) ∈ S, since
S is transitive). Prove that d(a, c) ≤ d(a, b) + d(b, c).
(b) Suppose (a, c) ∈ S and 0 < m < d(a, c). Prove that there is some
b ∈ A such that d(a, b) = m and d(b, c) = d(a, c) − m.
∗ 10. Suppose R is a relation on A and S is the transitive closure of R. For each
positive integer n, let J n ={0, 1, 2,..., n}.If a ∈ A and b ∈ A, we will
say that a function f : J n → A is an R-path from a to b of length n if
f (0) = a, f (n) = b, and for all i < n,( f (i), f (i + 1)) ∈ R.
n
(a) Prove that for all n ∈ Z , R ={(a, b) ∈ A × A | there is an R-path
+
from a to b of length n}.
(b) Prove that S ={(a, b) ∈ A × A | there is an R-path from a to b}.
11. Suppose R is a relation on A and S is the transitive closure of R.If f
is an R-path, then we say that the path is simple if f is one-to-one. (See
exercise 10 for the definition of R-path.)
n
+
(a) Prove that for all n ∈ Z , R \ i A ⊆{(a, b) ∈ A × A | there is a sim-
ple R-path from a to b of length at most n}.