Also, recall that the n
th
composition power of a relation on a
set is defined recursively by R
1
= R, and for each n
ε
,i
f
n
≥
1, then R
n+1
=R
n
R.
Prove that if R is a reflexive relation on a nonempty set A, then
R
⊆
R
n
for every positive integer n.
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7. (15 pts.)
(a) How many vertices does a tree with 37 edges
have?
(b)
What is the maximum number of leaves that a binary tree of
height 6 can have?
(c)
If a full 3ary tree has 24 internal vertices, how many
leaves does it have?
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8. (10 pts.)
Suppose that A is the set consisting of all real
valued functions with domain consisting of the interval [1,1] .
Let R be the relation on the set A defined as follows:
R = { (f,g)
(
∃
C)( C
ε
and f(0)  g(0) = C) }
Prove that R is an equivalence relation on the set A.
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 Spring '08
 STAFF
 Graph Theory, Binary relation, Tree traversal, Nested set model

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