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# Discrete Math : Probability, Functional Relations, Partitions and Primary Keys

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Name ________________________________ SSN __________________
CMSC 203 - Homework Assignment 4 - Due December 9, 2003
1. (a) Suppose I have a cooler full of cans of Coke, Pepsi, Sprite, Mountain Dew, Dr. Pepper, and
Slice sodas. How many distinct ways can I line these up on a table if the cooler contains 5 of each
type of soda?
(b) Suppose I have a large collection of coins consisting of Pennies, Nickels, Dimes, Quarters,
Half-Dollars, and Dollars. How many ways can I select distinct combinations of 50 coins if I must
have 5 of each type in each collection?
Name ________________________________ SSN __________________
CMSC 203 - Homework Assignment 4 - Due December 9, 2003
2. Consider the following sets with corresponding number of elements indicated in each region:
(a) Find P(A)
(b) Find P(A | (B Ç C))
A B
C
5 3 8
7
2
6 4
1 U
Name ________________________________ SSN __________________
CMSC 203 - Homework Assignment 4 - Due December 9, 2003
3. (a) Draw the directed graph of the relation R on A = {1, 2, 3, 4, 5, 6, 7, 8} defined as
R = {(a,b) | a,b Î A and a º b mod 3}.
2 3
· ·
1· · 4
8· · 5
· ·
7 6
(b) Find the matrix representing the relation on {1, 2, 3, 4, 5} given by:
R = {(1,3),(1,5),(2,2),(3,2),(3,4),(4,1),(4,3),(4,5),(5,2),(5,3),(5,4),(5,5)}
(c) Find MR o MS for the relations on whose matrix
representations are MR = and MS = .
1 0 0 0 0 1
0 0 1 0 1 1
0 0 1 1 0 0
0 0 0 1 0 0
1 1 1 0 0 0
0 1 0 1 0 1
0 0 1 0 1 0
0 1 0 1 1 0
1 1 0 1 1 0
1 0 1 1 1 0
1 0 0 1 0 0
0 1 0 1 0 0
Name ________________________________ SSN __________________
CMSC 203 - Homework Assignment 4 - Due December 9, 2003
4. Consider the relation, R, on the set A = {a, b, c, d, e, f, g, h} given by the graph:
(a) Find [d]
(b) Find the partition of A induced by R
a
c e
g
h
f
d
b
Name ________________________________ SSN __________________
CMSC 203 - Homework Assignment 4 - Due December 9, 2003
5. Let F be a function on the integers given by F(n) = n2+ 1.
(a) Show that the relation R = {(x,y) | x,y are integers and F(x) = F(y)}is a Reflexive, Symmetric,
and Transitive relation.
(b) Describe the partition of the integers induced by R.
.
Name ________________________________ SSN __________________
CMSC 203 - Homework Assignment 4 - Due December 9, 2003
6. Consider the database consisting of the following Fields and Records:
First Name Last Name Age Phone Height (in.) Weight
Michael Jones 26 555-1234 68 155
Mary Smith 31 555-4321 65 128
Ted Green 22 555-6789 74 210
Susan Green 20 555-6789 69 144
William Peters 44 555-9876 73 185
Alan Green 44 555-6789 70 185
(a) For this database, which Fields would serve as Primary Keys?
(b) Find P1,4,5
(c) Find P2,4