ab
\
\
\
cd
6. (10 pts.)
Provide a proof by induction that 2
n
≥
2n for
every positive integer n. Be explicit concerning your use of the
induction hypothesis in the inductions step.
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7. (15 pts.)
(a) How many edges does a tree with 37 vertices
have?
(b)
What is the maximum number of leaves that a binary tree of
height 10 can have?
(c)
If a full 3ary tree has 24 internal vertices, how many
vertices does it have?
8. (10 pts.)
Suppose that R is an equivalence relation on a
nonempty set A.
Recall that for each a
ε
A, the equivalence
class of a is the set [a] = {s  (a,s)
ε
R}. Prove the following
proposition:
If (a,b)
ε
R, then [a] = [b].
Hint: The issue is the set equality, [a] = [b], under the
hypothesis that (a,b)
ε
R. So pretend (a,b)
ε
R and use this to
show s
ε
[a]
→
s
ε
[b], and s
ε
[b]
→
s
ε
[a].
Be explicit
regarding your use of the relational properties of R.
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 Spring '08
 STAFF
 Graph Theory, pts, Binary relation, Tree traversal, Nested set model

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