MATH 239 ASSIGNMENT 1
Due Friday, 14th May at 11:45 a.m.
1. Let
S
n
denote the set of sequences of 0’s and 1’s of length
n
, written as vectors. For
example,
S
2
=
{
(0
,
0)
,
(0
,
1)
,
(1
,
0)
,
(1
,
1)
}
. Give a combinatorial proof that, for all
n
≥
1,
the number of sequences in
S
n
having an odd number of 1’s is equal to the number of
subsets of
{
1
,...,n
}
with an odd number of elements.
2. Let
n
be a positive integer. If
T
is a subset of
{
1
,...,n
}
, we say that
T
is an
n
palindrome
if, for every
i
∈ {
1
,...,n
}
, if
i
∈
T
then
n
+ 1

i
∈
T
. For instance,
{
1
,
2
,
4
,
5
}
is a
5palindrome, but is not a 6palindrome. Examples of 6palindromes include
{
2
,
5
}
and
the empty set. Show with a combinatorial proof that the number of
n
palindromes equals
2
n/
2
for
n
even, and 2
(
n
+1)
/
2
for
n
odd. (Hint: A set of cardinality
k
has 2
k
subsets.)
3. For
i
≥
1 let
U
i
be the set of ordered pairs of positive integers such that the larger one is
equal to
i
. For example,
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 Spring '09
 M.PEI
 Math, Set Theory, Vectors, Natural number

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