Discrete Structures
Aug 30 2011
Assignment 2:
Due at beginning of class Mon, Sept 12
Prof. Hopcroft
*******Note: this is the ﬁrst homework that will be subject to the style guide on the
website. Points will be deducted for homeworks not following the guidelines.********
1 Regular Expressions
Write a regular expression for the set of all strings of 0’s and 1’s that have two blocks of 1’s. A sample string
your regular expression should represent:
000000111111000111000
1.1 Answer
+4
for correct notation and regular expression,
+2
for each correct element of the regular expression,

1
for
extra 0’s and 1’s in an otherwise correct solution.
{
0*11*00*11*0*
}
2 Recurrence Relation
Find a close form solution for the following recurrence equations: (Also, show that you checked the ﬁrst four
values of your solution)
Saved for next week.
3 Induction
+5
for correct base case,
+5
for assuming
g
(
n
)
is true in a correct way,
+5
for manipulating
g
(
n
+ 1)
, and
+5
for overall correctness
Prove by induction that
g
(
n
) = 2
n
4
+ 2
n
2
+ 4 is divisible by 4 for all
n
≥
1.
Proof. Base Case
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 '07
 SELMAN
 Natural number, Rational number, Countable set, Georg Cantor, ﬁnite length strings

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