IST 230, Fall 2000
Exam I, In-class Part (60 points)
Name
Instruction: Most of the following problems require only short answers. No
elaborate supporting work is required.
Part 1 Basic Math (42 points)
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Part 2 Applications (18 points)
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Part 3 Extra Credit (5 points)
Problem 15
Problem16
Total
Part 1 Basic Math
1. Given two sets X  2, 5, 8 and Y  5, 6, 7, 8.
a) Find X  Y .
b) Find X  Y .
c) What is the set X  Y called?
d) What is X  Y ?
e) What is the set X Y called?
f) What is X Y ?
g) How many elements are there in the power set P(Y)?
h) How many 2-element subsets of Y are there?
2. (0, 1, 1, 0, 1, 1) and (0, 1, 0, 1, 1, 0)  Z 26 are two binary vectors of length 6.
a) Which two subsets of 1, 2, 3, 4, 5, 6 do the vectors represent?
b) Calculate the sum of these two vectors using the binary addition.
3. Give the edge list of the following directed graph. List the edges in lexicographic
order.
1
2
5
3
4
4. If a degree sequence of a graph is (1, 1, 1, 2, 2, 3, 4), what does the “Handshake
Lemma” tell you about this graph?
5. If a graph has the degree sequence (1, 0, 2, 1), is it connected?
6. Draw an unlabeled tree with the degree sequence (1, 1, 1, 1, 3, 3).
7. According to Cayley’s Theorem, how many labeled 5-node trees are there?


8. A  4, 5, 6, 7. List all elements of the relation R  (a, b)  A2 : a  1  b in
lexicographic order.
Part 2 Applications
9. Binary [7,4] Hamming codes are generated with the aid of a three-circle Venn
diagram.
a) Draw the Venn diagram, label each region by number 1, 2, …, 7, respectively.
b) Encode the binary string 0110.
c) Decode 1011101. [If there is an error in the code, correct it!]
10. The CD technology uses Eight-to-Fourteen Modulation to translate bytes into
codewords. Codes burnt on CDs are binary strings having at least two zeros between
any two ones. It also cannot have more than ten consecutive zeros.
a) Give one example of such a fourteen-bit codeword.
b) Let a k stands for the number of k-bits strings satisfying the requirement that
there are at least two zeros between and two ones. This is a Fibonacci numberlike sequence. For k  3 , a k  a k 1  a k 3 .
a 0  1 , a1  2 , a2  3 , a3  4 , a4  6 , a5  9 , a6  13
Find
a7 
a8 
11. In the discussion of public key encryption, what mathematical concepts are involved?
Circle two of the following concepts.
Binary strings
Fibonacci numbers
Large prime numbers
Modular arithmetic
Permutations
Binomial coefficients
12. The mesh topology used in networks is based on which type of graphs?
13. The star topology used in networks is based on which type of graphs?
14. The unit used in multi-dimensional database is called a “cell.” Which of the
following concepts are related to a cell? Circle two answers.
2
, K 8 , K 64 , K 8,8 , 8-cube, 64-cube
Z 28 , Z 264 , Z 64 , Z 64
Part 3 Extra Credit
15. A Hamiltonian cycle of the 3-cube is marked as follows. Following the path starting
at the node labeled 000, write up a non-standard Gray code.
111
110
101
011
100
010
001
000
16. You have to take some socks out of a completely dark room. There are a dozen pairs
of socks: Six pairs are white and six pairs are black. You can not tell which sock is
which kind when you are in the room. You can take out the room as many socks as
you want, as long as there is at least a pair of matching socks. What is the least
number of socks you need to take out of the room? Answer and explain. [Hint: Use
pigeon hole principle.]