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Math 184A
First Exam (40 points)
19 October 2005
•
Please put your name and ID number on your blue book.
•
The exam is CLOSED BOOK except for one page of notes.
•
Calculators are NOT allowed.
•
You must show your work to receive credit.
1. (10 pts.) Prove that the number of ordered lists without repeats (including the empty
list) that can be constructed from an
n
set is nearly
n
!
e
.
Hint
: By Taylor’s theorem,
e
is nearly 1 + 1
/
1! + 1
/
2! + 1
/
3! +
· · ·
+ 1
/n
!.
2. (10 pts.) For each of the following,
EITHER give an example of the thing that is described
OR explain why none exists.
(a) A surjection from
{
1
,
2
,
3
}
to
{
a, b, c, d
}
.
(b) An injection from
{
1
,
2
,
3
}
to
{
a, b, c, d
}
.
(c) A permutation
f
of
{
1
,
2
,
3
,
4
,
5
}
such that
f
40
is NOT the identity function and
f
40
n
=
f
. Also, if you ±nd such an
f
, compute
f
40
.
Remember
that the identity function is the function
g
such that
g
(
x
) =
x
for all
x
and
f
40
(
x
) is
f
(
f
(
· · ·
f
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This note was uploaded on 06/18/2009 for the course MATH 184a taught by Professor Staff during the Fall '08 term at UCSD.
 Fall '08
 staff
 Math

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