Mathematics Study Material: Bit Operations, Graph Theory, and Algebra - Prof. Pantheris | Study notes Discrete Mathematics

Table Of Content 
Table of Contents 
Learning Outcome 01 .............................................................................................................. 5 
Part – 1............................................................................................................................... 5 
1. Let U = {l, 2, 3, 4, 5, 6, 7, 8, 9, and 10} be a universal set. Let A, B, C such that A= {l, 3, 4, 
8}, B = {2, 3, 4, 5, 9, 10}, and C = {3, 5, 7, 9, 10}. Use bit representations for A, B, and C 
together with UNION, INTER, DIFF, and COMP to find the bit representation for the 
following: ............................................................................................................................ 5 
Part - 2 ............................................................................................................................... 6 
2.1 Write the multisets of prime factors of given numbers. .................................................... 6 
2.2 Write the multiplicities of each element of multisets in part 2(1-I, ii, iii) separately. ......... 6 
2.3 Find the cardinalities of each multiset in part 2-1. ............................................................ 7 
Part – 3............................................................................................................................... 8 
3. A student has 37 days to prepare for an exam. From past experience, he knows that he will 
need no more than 60 hours of study. To keep from forgetting the material, he wants to study 
for at least one hour each day. Show that there is a sequence of successive days during which 
he will have studied exactly 13 hours.................................................................................... 8 
3.1 A certain computer system breaks down in two mains. Ways: faults on the network and 
power supply faults. Of the last 50 breakdowns,42 involved network faults and 20 power 
failures. In 13 cases, both the power supply and the network were faulty. How many 
breakdowns were attributable to other kinds of failure? ......................................................... 9 
Part - 4 ............................................................................................................................. 10 
4. ...................................................................................................................................... 10 
i. Write a function that represents the total monthly cost C(x) of Mary Ann’s cell ................ 10 
ii. Find the inverse function ................................................................................................ 10 
iii. What do x and C-1(x) represent in the context of the inverse function? ........................... 10 
iv. How many additional minutes did Mary Ann use if her bill for her third month was $48.89?
 ......................................................................................................................................... 10 
4.1. An online music subscription service allows members to download songs for $0.99 each 
after Paying a monthly service charge of $3.99. The total monthly cost C(x) of the service in 
dollars is C(x)=3.99 +0.99x, where x is the number of songs downloaded. ........................... 11 
i. Find the inverse function ................................................................................................. 11 
ii. What do x and C-1(x) represent in the context of the inverse function? ............................ 11 
iii. How many songs were downloaded if a member’s monthly bill is $27.75? ..................... 11 
Learning Outcome 02 ............................................................................................................ 12 

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Table Of Content

Learning Outcome 01 .............................................................................................................. 5 Part – 1 ............................................................................................................................... 5

Let U = {l, 2, 3, 4, 5, 6, 7, 8, 9, and 10} be a universal set. Let A, B, C such that A= {l, 3, 4, 8}, B = {2, 3, 4, 5, 9, 10}, and C = {3, 5, 7, 9, 10}. Use bit representations for A, B, and C together with UNION, INTER, DIFF, and COMP to find the bit representation for the following: ............................................................................................................................ 5 Part - 2 ............................................................................................................................... 6 2.1 Write the multisets of prime factors of given numbers. .................................................... 6 2.2 Write the multiplicities of each element of multisets in part 2(1-I, ii, iii) separately. ......... 6 2.3 Find the cardinalities of each multiset in part 2-1. ............................................................ 7 Part – 3 ............................................................................................................................... 8
A student has 37 days to prepare for an exam. From past experience, he knows that he will need no more than 60 hours of study. To keep from forgetting the material, he wants to study for at least one hour each day. Show that there is a sequence of successive days during which he will have studied exactly 13 hours.................................................................................... 8 3.1 A certain computer system breaks down in two mains. Ways: faults on the network and power supply faults. Of the last 50 breakdowns,42 involved network faults and 20 power failures. In 13 cases, both the power supply and the network were faulty. How many breakdowns were attributable to other kinds of failure? ......................................................... 9 Part - 4 ............................................................................................................................. 10
...................................................................................................................................... 10 i. Write a function that represents the total monthly cost C(x) of Mary Ann’s cell ................ 10 ii. Find the inverse function ................................................................................................ 10 iii. What do x and C-1(x) represent in the context of the inverse function? ........................... 10 iv. How many additional minutes did Mary Ann use if her bill for her third month was $48.89? ......................................................................................................................................... 10 4.1. An online music subscription service allows members to download songs for $0.99 each after Paying a monthly service charge of $3.99. The total monthly cost C(x) of the service in dollars is C(x)=3.99 +0.99x, where x is the number of songs downloaded. ........................... 11 i. Find the inverse function ................................................................................................. 11 ii. What do x and C-1(x) represent in the context of the inverse function? ............................ 11 iii. How many songs were downloaded if a member’s monthly bill is $27.75? ..................... 11 Learning Outcome 02 ............................................................................................................ 12

Part - 1 ............................................................................................................................. 12

Model the following situations as (possibly weighted, possibly directed) graphs. Draw each graph, and give the corresponding adjacency matrices......................................................... 12 Part- 2 ............................................................................................................................... 12 a) Draw a graph G to represent this situation. ...................................................................... 12 b) List the vertex set, and the edge set, using set notation. In other words, show sets V and E for the vertices and edges, respectively, in G = {V, E}. ....................................................... 12 c) Draw an adjacency matrix for G. .................................................................................... 13
Find three distinct Hamiltonian cycles in the following graph. Also find their weights...... 13 Part- 3 ............................................................................................................................... 14
State the Dijkstra’s algorithm for a directed weighted graph with all non-negative edge weights. ............................................................................................................................. 14
Find the shortest path spanning tree for the weighted directed graph with vertices A, B, C, D, and E given using Dijkstra’s algorithm........................................................................... 15 Part - 4 .................................................................................................................................. 16 Construct a proof for the five-color theorem for every planar graph. .................................... 16
Discuss how efficiently Graph Theory can be used in a route planning project for a vacation trip from Colombo to Trincomalee by considering most of the practical situations (such as millage of the vehicle, etc.) as much as you can. Essentially consider the two-fold ............... 17 Learning Outcome 03 ............................................................................................................ 18
Use Boolean algebra to simplify the following expression, then draw a logic gate circuit for the simplified expression:................................................................................................... 18 Part – 2 to Part 3. ............................................................................................................. 19 Part 4 ............................................................................................................................... 20
1. Prove the following equations using the Boolean algebraic theorem............................. 20 4.2 Home Alarm System which informs the house owner of a theft at home. You need to design using OR gates. This circuit protects two windows, the front and back door of the house. When any window or door is opened, an alarm system will be on. The logic gates which is connected to windows and doors are connected to sensors. Input is "1" when windows or doors are open; input is 0 when windows or door are closed. Draw a truth table with a circuit diagram. ....................................................................................................... 21 Learning Outcomes 04 ........................................................................................................... 22 Part 1 ............................................................................................................................... 22
Describe the characteristics of different binary operations that are performed on the same set. .................................................................................................................................... 22
Justify whether the given operations on relevant sets are binary operations or not............. 22

Mathematics Study Material: Bit Operations, Graph Theory, and Algebra - Prof. Pantheris, Study notes of Discrete Mathematics

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Partial preview of the text

Download Mathematics Study Material: Bit Operations, Graph Theory, and Algebra - Prof. Pantheris and more Study notes Discrete Mathematics in PDF only on Docsity!

Table of Contents

Learning Outcome 01

Part – 1

1. Let U = {l, 2, 3, 4, 5, 6, 7, 8, 9, and 10} be a universal set. Let A, B, C such that

A= {l, 3, 4, 8}, B = {2, 3, 4, 5, 9, 10}, and C = {3, 5, 7, 9, 10}. Use bit

representations for A, B, and C together with UNION, INTER, DIFF, and COMP

to find the bit representation for the following:

2.3 Find the cardinalities of each multiset in part 2-1.

Part – 3

3. A student has 37 days to prepare for an exam. From past experience, he knows

that he will need no more than 60 hours of study. To keep from forgetting the

material, he wants to study for at least one hour each day. Show that there is a

sequence of successive days during which he will have studied exactly 13 hours.

Part - 4

i. Write a function that represents the total monthly cost C(x) of Mary Ann’s cell

ii. Find the inverse function

iii. What do x and C-1(x) represent in the context of the inverse function?

iv. How many additional minutes did Mary Ann use if her bill for her third month

was $48.89?

4.1. An online music subscription service allows members to download songs for

$0.99 each after Paying a monthly service charge of $3.99. The total monthly cost

C(x) of the service in dollars is C(x)=3.99 +0.99x, where x is the number of songs

downloaded.

i. Find the inverse function

ii. What do x and C-1(x) represent in the context of the inverse function?

iii. How many songs were downloaded if a member’s monthly bill is $27.75?

c) Draw an adjacency matrix for G.

3. Find three distinct Hamiltonian cycles in the following graph. Also find their

weights.

Part- 3

1. State the Dijkstra’s algorithm for a directed weighted graph with all non-

negative edge weights.

Part - 4

Construct a proof for the five-color theorem for every planar graph.

2. Discuss how efficiently Graph Theory can be used in a route planning project

for a vacation trip from Colombo to Trincomalee by considering most of the

practical situations (such as millage of the vehicle, etc.) as much as you can.

Essentially consider the two-fold

Part – 2 to Part 3.

Part 4

4. 1) Prove the following equations using the Boolean algebraic theorem.