MATH 239 Assignment 3
This assignment is due at noon on Friday, June 12, 2009, in the drop boxes opposite the
Tutorial Centre, MC 4067.
1. Let
a
and
b
be two distinct nonempty binary strings, and let
A
=
{
a,b
}
. Determine
whether the following two statements are true or false, and prove your assertions.
(a) If
a
and
b
have the same length, then strings of
A
*
are uniquely created.
(b) If
a
and
b
have different lengths, then strings of
A
*
are uniquely created.
2. For each of the three parts in this quesion,
i. ﬁnd a decomposition that uniquely creates elements of
S
; and
ii. show that the generating function for
S
with respect to length is the given
rational function.
(a)
i.
S
is the set of binary strings that do not contain “1000” as a substring.
ii.
Φ
S
(
x
) =
1
1

2
x
+
x
4
.
(b)
i.
S
is the set of binary strings where each block has odd length.
ii.
Φ
S
(
x
) =
1 +
x

x
2
1

x

x
2
.
(c)
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 Spring '09
 M.PEI
 Math, Combinatorics, Rational function, CN, Recurrence relation, Generating function, binary strings

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