Discrete Mathematics
Lecture 1
Dr.-Ing. Erwin Sitompul
President University
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Discrete Mathematics
Text Book and Syllabus
 Text book:
Kenneth H. Rosen, “Discrete Mathematics and Its
Applications”, 6th Edition, McGraw-Hill International
Edition, 2007.
 Tentative Syllabus:
1. Logic and Proofs
2. Sets
3. Relation and Functions
4. Sequences and Summations
5. Number Theory
6. Counting and Combinatorial
7. Graphs
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Discrete Mathematics
Grade Policy
Final Grade =
5% Notes + 9% Homework + 19% Quizzes +
29% Midterm Exam + 39% Final Exam +
Extra Points
 Notes must be handwritten, and will be graded based on the completeness
and clarity.
 The handwritten note will be checked on Lecture 11 (after Quiz 3), and
given back to you on Lecture 12. It contributes 5% of final grade.
 Homeworks will be given in fairly regular basis. The average of homework
grades contributes 9% of final grade.
 Written homeworks are to be submitted on A4 papers, otherwise they will
not be graded.
 Homeworks must be submitted on time, one day before the next lecture.
Late submission will be penalized by point deduction of –10·n, where n is
the total number of lateness made.
 There will be 3 quizzes. Only the best 2 will be counted. The average of quiz
grades contributes 19% of final grade.
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Discrete Mathematics
Grade Policy
 Midterm and final exams follow the schedule released by AAB (Academic
Administration Bureau).
 Make up of quizzes must be requested within one week after the schedule
of the respective quiz.
 Make up for mid exam and final exam must be requested directly to AAB.
Thermal Physics
Homework 2
Emelie Raturandang
002202400058
21 March 2027
No.1. Answer: . . . . . . . .
● Heading of Written
Homework Papers (Required)
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Discrete Mathematics
Grade Policy
 Extra points will be given if you solve a problem in front of the class. You will
earn 1 or 2.
 Lecture slides can be copied during class session. It is also available on
internet. Please check the course homepage regularly.
http://zitompul.wordpress.com
 The use of internet for any purpose during class sessions is strictly
forbidden.
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Discrete Mathematics
Chapter 0
Introduction
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Chapter 0 Introduction
What is “Discrete Mathematics”?
 Discrete Mathematics is a branch of mathematics that discuss about
discrete objects or structures. Discrete objects or structures can assume
only distinct, separated values.
 The term “discrete mathematics” is therefore used in contrast with
“continuous mathematics” which is the branch of mathematics dealing with
objects that can vary smoothly (for example, calculus).
 What is the definition of discrete?
 An object can be said to be discrete if :
 It consists of unconnected distinct parts/members
 It consists of finite or countable parts/members
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Chapter 0 Introduction
What is “Discrete Mathematics”?
 Example of discrete objects:
 Integers: between two integers there is no other integer.
 Propositions: either true or false, there are no half true or half false.
 Sets: an object is whether in a set or not in a set, it never partly in and
partly out.
 Relations: a pair of objects are whether related or not at all.
 Graphs: In a network, between two terminals of a direct connection,
there are no other terminals.
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Chapter 0 Introduction
Discrete Mathematics in Electrical Engineering
 Discrete Mathematics is relevant to Electrical Engineering, because we
often deal with objects with discrete properties.
 Consider a digital signal processing of a video clip:
 The color is discretized into 3 hues: Red, Green, and Blue.
 The hue is discretized into discrete intensity level, 0-255.
 The pixels are discrete objects in space.
 The frames are discrete objects in time.
 In the digital world nowadays, often discrete variables are used to represent
a phenomena in discretely rather than continuously.
 The states of a computer program are discrete.
 The states of a digital hardware design are discrete.
 Discrete structures (sets, functions, relations, trees, graphs) are very
useful for representing data in computers.
 Connecting a generator or a load to a power grid is a discrete change.
 The measurement result of a sensor is discretized before it is processed.
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Chapter 0 Introduction
What is “Discrete Mathematics”?
 Some examples of problems related with Discrete Mathematics:
 How many different password can be made out of 8 different
characters?
 How does a credit card number is validated?
 How many 8-bit-long binary string combinations can be made if the
sum of bit-1 must be odd?
 How to determine the shortest path between point A and point B in a
factory complex?
 Proof that a combination of 3s and 5s can result any integer number
higher than 8.
 How many different FM frequencies are needed to be assigned to
commercial radio stations so that the stations do not interfere each
other?
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Chapter 0 Introduction
What is “Discrete Mathematics”?
 Some examples of problems related with Discrete Mathematics:
 How to construct logic circuit for a seven segments?
 Can you walk through all the streets in your housing complex exactly
once and come back to the original position?
 “Cheap food is not tasty.”
“Tasty food is not cheap.”
Are both statements telling us the same thing?
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Discrete Mathematics
Chapter 1
Logic and Proofs
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Chapter 1 Logic and Proofs
Logic and Proposition
 Logic defines a formal language for representing knowledge and for making
logical inferences.
 Logic helps us to understand how to construct a valid argument.
 Logic defines:
 Syntax of statements/propositions
 The meaning of statements/propositions
 The rules of logical inference
 Logic is the basic of reasoning, which is based on relations between
statements/propositions.
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Chapter 1 Logic and Proofs
Proposition
 A proposition is a statement that can be either true or false, but not both.
 Some examples of proposition:
 “13 is an odd number.”
 “Ir. Soekarno was graduated from UGM.”
 “It is raining today.”
 “The day after tomorrow is Wednesday.”
 “There are other life forms on other planets in the universe.”
 “For any integer n  0, there exists 2n which is an even number.”
 “x + y = y + x for any real number x and y.”
 “1 + 1 = 2.”
 “x – 5 = 8, where x = 14.”
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Chapter 1 Logic and Proofs
Proposition
 These statements are not proposition:
 “How old are you?”
·A question is not a proposition
 “Do the quiz without cooperating!”
·A command is not a proposition
 “x + 5 = 3.”
·Since x is not specified, neither true or false
 “y > 5.”
·Since y is not specified, neither true or false
 “He is tall.”
·Since ‘he’ is not specified, neither true or false
 Conclusion: Propositions are declarative sentences or close sentence.
 If a proposition is made out of mathematical equations, then the equations
must possess a solution/answer so that its truth value can be determined.
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Chapter 1 Logic and Proofs
Proposition
 Propositions are denoted with lowercase letters, starting with p, such as p,
q, r, s, …
 Example of proposition:
 p : “13 is an odd number.”
 q : “Ir. Soekarno was graduated from UGM.”
 r : “2 + 2 = 4.”
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Chapter 1 Logic and Proofs
Compound Proposition
 A compound proposition can be built from elementary propositions by
using logical connectives.
 Logical connectives are:
 Negation
 Conjunction
 Disjunction
 Exclusive or
 Conditional (Implication)
 Biconditional
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Chapter 1 Logic and Proofs
Compound Proposition
 Given propositions p and q, then:
1. Negation
: not p
p
2. Conjunction : p and q
pq
3. Disjunction
: p or q
pq
¬p
 Combination of more than one proposition result in a compound
proposition.
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Chapter 1 Logic and Proofs
Compound Proposition
 Example: The following propositions are given:
p : Today is rainy.
q : The class is cancelled.
The two propositions can be combined to become:
pq
: “Today is rainy and the class is cancelled.”
pq
: “Today is rainy or the class is cancelled.”
p
: “It is not true that today is rainy.”
“It is not the case that today is rainy.”
“Today is not rainy.”
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Chapter 1 Logic and Proofs
Compound Proposition
 Example: Given the following propositions
p : “The man is competent.”
q : “The man is assertive.”
Express the following proposition combinations using symbolic notation.
i. “The man is competent and assertive.”
pq
ii. “The man is competent but not assertive.”
p  q
iii. “The man is neither competent nor assertive.”
p  q
iv. “It is not the case that the man is incompetent
(p  q)
or not assertive.”
v. “The man is competent, or incompetent and assertive.”
p(p  q)
vi. “That the man is incompetent as well as assertive,
(p  q)
is not true.”
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Chapter 1 Logic and Proofs
Truth Table of Compound Proposition
 A truth table displays the relationship between the truth values (T or F) of
different propositions for a given connective.
 The truth table of a compound proposition shows all possible combinations
of truth values of elementary propositions.
● Negation
● Conjunction
● Disjunction
p
p
p
q
pq
p
q
pq
T
F
T
T
T
T
T
T
F
T
T
F
F
T
F
T
F
T
F
F
T
T
F
F
F
F
F
F
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Chapter 1 Logic and Proofs
Some Examples
 Example:
p : “17 is a prime number.” T
q : “Prime number is always odd.”
F
p  q : “17 is a prime number and prime number is always odd.”
F
 Example:
F
T
“Surabaya was the capital city of Indonesia or tomorrow is Friday.”
T
 Example:
F
T
“It is not true that February is the shortest month and energy cannot be
created.”
F
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Chapter 1 Logic and Proofs
Constructing the Truth Table
 Example: Build the truth table of the proposition (p  q)  (q  r).
p
q
r
pq
q
q  r
(p  q)  (q  r)
T
T
T
T
F
F
T
T
T
F
T
F
F
T
T
F
T
F
T
T
T
T
F
F
F
T
F
F
F
T
T
F
F
F
F
F
T
F
F
F
F
F
F
F
T
F
T
T
T
F
F
F
F
T
F
F
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Chapter 1 Logic and Proofs
Tautology, Contradiction, and Contingency
 A compound proposition which is always true for all cases is called
tautology.
 A compound proposition which is always false for all cases is called
contradiction.
 A proposition that is neither a tautology nor contradiction is called a
contingency.
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Chapter 1 Logic and Proofs
Tautology, Contradiction, and Contingency
 Example: p  (p  q) is a tautology.
p
q
pq
(p  q)
p  (p  q)
T
T
T
F
T
T
F
F
T
T
F
T
F
T
T
F
F
F
T
T
 Example: (p  q)  (p  q) is a contradiction.
p
q
pq
pq
(p  q)
(p  q)  (p  q)
T
T
T
T
F
F
T
F
F
T
F
F
F
T
F
T
F
F
F
F
F
F
T
F
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Chapter 1 Logic and Proofs
Equivalence of Compound Propositions
 Two compound propositions A(p,q,…) and B(p,q,…) are said to be logically
equivalent if they have identical truth table.
 Notation: A(p,q,…)  B(p,q,…)
 Equivalent propositions are important for logical reasoning since they can
be substituted and can help us to make simplification, make logical
argument, infer new propositions,
 Example: De Morgan’s Law (p  q)  p  q.
p  q (p  q)
p
q
p  q
F
F
F
F
F
T
F
T
T
T
F
T
T
F
T
F
F
T
T
T
T
p
q
T
T
T
T
F
F
F
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Chapter 1 Logic and Proofs
Logical Equivalences
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Chapter 1 Logic and Proofs
Logical Equivalences
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Chapter 1 Logic and Proofs
Logical Equivalences
 Example: Show that p  (p  q) and p  q are logically equivalent.
p  (p  q )  p  (p  q)
(De Morgan’s Law)
 (p  p)  (p  q)
(Distributive Law)
 T  (p  q)
(Negation Law)
 p  q
(Identity Law)
 Example: Negate the following proposition with De Morgan’s Law
“The summer in Mexico is cold and sunny.”
(p  q)  p  q
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Chapter 1 Logic and Proofs
Logical Equivalences
 Example: Proof the truth of Absorption Law p  (p  q)  p .
p  (p  q)  (p  F)  (p  q)
(Identity Law)
 p  (F  q)
(Distributive Law)
 pF
(Domination Law)
 p
(Identity Law)
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Chapter 1 Logic and Proofs
Exclusive Disjunction
 Let p and q are propositions. The proposition “p exclusive or q”, denoted by
p  q, is true when exactly one of p and q is true and is false otherwise.
● Exclusive
Disjunction
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p
q
pq
T
T
F
T
F
T
F
T
T
F
F
F
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Chapter 1 Logic and Proofs
Conditional Proposition
 Conditional proposition is also called implication.
 Notation: p  q
 The way to pronounce: “if p, then q”
● Conditional
(Implication)
p
q
pq
T
T
T
T
F
F
F
T
T
F
F
T
 In p  q, the proposition p as called hypothesis or premise, and the
proposition q is called conclusion or consequence.
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Chapter 1 Logic and Proofs
Conditional Proposition
 Various ways to express implication p  q:
 If p, then q.
 If p, q.
 p implies/causes q.
 q if p.
 p only if q.
 p is the sufficient condition for q.
 p is sufficient for q.
 q is the necessary condition for p.
 q is necessary for p.
 q whenever p.
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Chapter 1 Logic and Proofs
Conditional Proposition
 Example:
F
T
“If Albert Einstein is still alive then man ever walked on the moon.”
 Example: T
F
“If now is 2016 then Indonesia is already independent for 80 years.”
T
F
 Example: (Truth value still cannot be determined)
 “If I pass the exams, then I will give you my notebook.”
 “If the temperature reaches 80 °C, then the alarm will buzz.”
 “If you have not enrolled properly, then your name will not be printed
on the attendance list.”
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Chapter 1 Logic and Proofs
Conditional Proposition
 Example: Implication in various forms
 “If today is rainy, then the flowers will grow well.”
 “If the gas pedal is pressed deeper, the car moves faster.”
 “Ice melted in north and south pole causes the increase of sea level.”
 “He is willing to go if he is given travel allowance.”
 “Andy can take Computer Organization only if he already passed
Discrete Mathematics.”
 “The sufficient condition for a gas station to explode is small cigarette
sparks.”
 “The necessary condition for Indonesia to win the World Cup is by
hiring a famous foreign trainer.”
● Convert all propositions
above into “If p then q.”
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Chapter 1 Logic and Proofs
Conditional Proposition
 Example:
Show that p  q is logically equivalent with p  q.
p
q
pq
p
p  q
T
T
T
F
T
T
F
F
F
F
F
T
T
T
T
F
F
T
T
T
“If p, then q”  “Not p or q”
 Example:
Determine the negation of p  q.
(p  q)  (p  q)
 (p)  q
 p  q
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Chapter 1 Logic and Proofs
Conditional Proposition
 Example: Express the following proposition combinations using symbolic
notation.
i. “You can access the internet from campus only if
a  (s  g)
you are a student or you are not a university guest.”
ii. “You cannot ride the roller coaster if you are under
(u  o)  r
100 cm tall unless you are older than 16 years old.”
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Chapter 1 Logic and Proofs
Conditional Proposition
 Example: Assume two elementary propositions,
p: “You drive over 100 km/h.”
q: “You get a speeding ticket.”
Translate each of these sentences to logic.
i. “You do not drive over 100 km/h.”
ii. “You will get a speeding ticket if you drive over 100 km/h.”
iii. “If you do not drive over 100 km/h then you will not get a
speeding ticket.”
iv. “Driving over 100 km/h is sufficient for getting a speeding ticket.”
v. “You get a speeding ticket, although you do not drive over 100 km/h.”
vi. “You drive over 100 km/h, but you do not get a speeding ticket.”
i. p
ii. p  q
iv. p  q
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iii. p  q
v. q  p
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vi. p  q
Discrete Mathematics 1/38
Chapter 1 Logic and Proofs
Application of Propositional Logic
 Inference and Reasoning: new true propositions are inferred from existing
ones.
 Used in the fields of:
 Artificial Intelligence  build programs that act intelligently, often rely
on symbolic manipulations.
 Rule based (expert) systems  encode knowledge about the world in
logic, new facts are inferred from existing facts following the semantics
of logic.
 Automatic theorem proof  encode existing knowledge (e.g. about
math) using logic and show that some hypothesis is true.
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Chapter 1 Logic and Proofs
Homework 1
1. Two merchants publish new marketing campaign to attract more
customers. The first merchant launches a motto “Good stuffs are not
cheap.” The second merchants says “Cheap stuffs is not good.”
a) Examine whether both mottos tell the same message or not.
b) In your opinion, which motto is better?
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Chapter 1 Logic and Proofs
Homework 1A
1. Examine the statements S1 and S2, and find out whether they are logically
equivalent or not.
S1: “Excellent physical condition is the necessary condition
to become a soldier.”
S2: “If someone does not have excellent physical condition,
then he cannot become a soldier.”
2. Show that p  q  q  p by using (i) logical equivalences; (ii) truth
table.
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