Posted: December 1st, 2015

Discrete maths

Discrete maths

Set 1
1.    The following series are all arithmetic. In each case, tell the next term, give a formula for the

nth term assuming that n = 1 for the first term, and give a formula for the kth term assuming that k = 0 for

the first term.
A.    16, 14, 12, 10, 8, 6, . . .
B.    -6, -8, -10, -12, -14 – 16, . . .
C.    0, 5, 10, 15, 20, 25, . . .
D.    0, 1/2 , 1, 3/2 , 2, 5/2 , . . .
E.    -72, -60, -48, -36, -24, -12, . . .

2.    The following series are all geometric. In each case, tell the next term, give a formula for the nth term assuming that n = 1 for the first term, and give a formula for the kth term assuming that k = 0 for the first term.
A.    -4, -16, -64, -256, -1024, . . .
B.    1/3 , 1/9 , 1/27 , 1/81 , 1/243 , . . .
C.    1/2 , – 1/4 , 1/8 , – 1/16 , 1/32 , . . .
D.    1, .5, .25, .125, .625, . . .
E.    7, -14, 28, -56, 112, . . .

3.    The following series are all quadratic. In each case, tell the next term, show the sequence of first

differences, show the constant second difference, give a formula for the n th term assuming that n = 1 for

the first term, and give a formula for the k th term assuming that k = 0 for the first term.
A.    1, 2, 5, 10, 17, . . .
B.    -1, -.5, .5, 2, 4, . . .
C.    -1, -5, -10, -16, -23, . . .
D.    4, 8, 11, 13, 14, . . .
E.    4, 0, 0, 4, 12, . . .

4.    Write each of the following sums using summation notation. Try to make you answers as simple as

possible.
A.    4 + 7 + 10 + 13 + • • • + 304
B.    1 + 2 + 4 + 8 + … + • • • + 1, 073, 741, 824
C.    1/5 – 1/25 + 1/125 – 1/625 + • • • + 1/30,517,578,125

5.    Apply Gauss’s trick to evaluate each of the following sums.
A.    4 + 7 + 10 + 13 + • • • + 304
B.    1 + 2 + 3 + 4 + • • • + 1000
C.    15 + 22 + 29 + 36 + • • • + 715

Set 2

Figure 1
1.    Use Figure 1 to answer the following questions.
a.    List all of the paths, with no repeated vertices, from vertex E to vertex C
b.    Give the degree of each vertex
c.    What is the length of the longest cycle in this graph?
d.    What is the length of the shortest cycle in this graph?

Figure 2
2.    Use Figure 2 to answer the following questions.
a.    List all of the paths, with no repeated vertices, from vertex E to vertex C
b.    Give the degree of each vertex
c.    What is the length of the longest cycle in this graph?
d.    What is the length of the shortest cycle in this graph?

Figure 3
3.    Use Figure 3 to answer the following questions.
a.    What is the weight of the path (A,B,D,E)
b.    How many paths of length 3 or less are there from A to D? What is the weight of each of these paths?
c.    What is the length of the longest simple path from vertex A to vertex E? Is there more than one path of this length?
d.    What is the shortest path from vertex A to vertex D? Which path from vertex A to vertex D has the lowest weight?

Figure 4
4.    Use Figure 4 to answer the following questions
a.    What is the weight of the path (A,B,D,E)
b.    How many paths of length 4 or less are there from A to E? What is the weight of each of these paths?
c.    What is the length of the longest simple path from vertex A to vertex E? Is there more than one path

of this length?
d.    What is the shortest path from vertex A to vertex D? Which path from vertex A to vertex D has the

lowest weight?
Instructor: Joshua L. Smith

Figure 1
5.    Give the adjacency list for the graph in Figure 1

Figure 2
6.    Give the adjacency list for the graph in Figure 2

7.    Draw the graph described by this adjacency list.

8.    Give the adjacency matrix for the graph in Figure 1
A    B    C    D    E
A
B
C
D
E

9.    Give the adjacency matrix for the graph in Figure 2.
A    B    C    D    E    F
A
B
C
D
E
F

10.    Find the relation in each of the following statements and determine if it is reflexive, symmetric or

transitive.
a.    6 and 2 divides 12
b.    3 and 6 do not divide 8
c.    The set A intersects set B at points {1,3,4,6}
d.    The set C disjoins set D at points {2,3,5}
e.    B is congruent modulo n to A
f.    D is not congruent modulo n to C
g.    John is the third-cousin twice removed of James
h.    James is the brother of Steve
i.    Kelly is the roommate of Ashley
j.    Billy is Tom’s neighbor
k.    Shelby is a student at the University of Illinois Springfield
l.    Andrew is a patient at St. John’s Hospital
Set 3
Problem 1.
According to the traditional song, on the first day of Christmas (25th December), my true love sent to me:
A partridge in a pear tree
On the second day of Christmas (26th December), my true love sent to me THREE presents:
Two turtle doves
A partridge in a pear tree

On the third day of Christmas (27th December and so on) my true love sent to me SIX presents:
Three French hens
Two turtle doves
A partridge in a pear tree

This carries on until the twelfth day of Christmas, when my true love sends me:
Twelve drummers drumming
Eleven pipers piping
Ten lords a-leaping
Nine ladies dancing
Eight maids a-milking
Seven swans a-swimming
Six geese a-laying
Five gold rings
Four calling birds
Three French hens
Two turtle doves
A partridge in a pear tree

After the twelve days of Christmas are over, how many presents has my true love sent me altogether?

Problem 2.
Given only one of each letter in the alphabet, what are the smallest and largest number that you can write down? Remember, you are only allowed to use one of each letter in the alphabet!

Problem 3.
You have a basket containing 5 oranges. You have 5 friends, who each desire an orange. You give each of your friends one orange.
Now, all of your friends have one orange each, yet there is an orange remaining in the basket.
How?
Problem 4.
You are at an unmarked intersection… one way is to the City of Lies and the other is to the City of Truth.
Citizens of the City of Lies always lie and the citizens of the City of Truth always tell the truth.
A citizen of one of those cities (you don’t know which one) is at the intersection. What question could you ask them to find the way to the City of Truth?

Problem 5.
Four people are traveling to different places on different types of transport. Their names are Rachel, John, Steve and Cindy. They either went on a train, car, plane or ship.
Steve hates flying
Cindy has to rent her vehicle
John gets seasick

Name one of the 3 possible ways these people can travel.

Problem 6.
I have three special dice that are four-sided. They have one letter on each side. When I roll them together I get three random letters which I try to rearrange into a word. In my eight tries so far I have made the following words.

CAT, SON, POD, RIG, PEG, TAP, DIN, APE
What are the letters on each dice?

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