# 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

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