Topic 0. Introduction
Aims and objectives of the subject “Discrete
mathematics”:
 To introduce students to some main concepts and methods of discrete
mathematics (sets, relations, elements of mathematical logic, Boolean
algebra, graphs, combinatorial analysis, etc.).
 To make students aware of some applications of discrete mathematics in
the area of discrete information processing by computers.
 To teach students how to apply concepts and methods of discrete
mathematics for solving their information processing problems (to be
able to analyze the structure of discrete components and their relation in
the problem and to design an algorithm that produces the required data).
About the word “discrete”[dis'kri:t]:
In every day English, the word “discrete” means separate, distinct. A discrete
element is easily distinguished from other elements in a group (set). It is
recognized clearly.
In a continuous set, it’s not easy to say which value is represented by a given
element.
For example, in a set of real numbers, element 3.14 may represent an infinite
number of rational and irrational numbers depending on the digits that follow
the digit “4”. A discrete element has only one unambiguous value.
There is another word “discreet”[dis'kri:t] that is pronounced in the same
way but has a different meaning (of people behavior): careful and sensible in
what one chooses not to say, not causing discomfort. Ex. It was not very
discreet of you to call me up at the office.
1
What is the difference between the discrete
mathematics and other mathematical courses?
Discrete mathematics studies finite sets and countable infinite sets. Note: In a
countable set, each element can be assigned a distinct integer number. It also
𝟏
does not use transition to the limit value like 𝐥𝐢𝐦 = 𝟎.
𝒏→∞ 𝒏
On the other hand, in the course of calculus, almost all sets under consideration
are infinite, like the set of rational numbers, the set of all values of a function,
etc.
Sets studied in discrete mathematics are usually finite. But very often such finite
sets contain a very large number of elements. For example, the set of all
10
Boolean functions of 10 arguments contains 22 elements. This is a huge
number. A great number of elements in many sets makes it necessary to develop
special methods for specification, research and analysis of such sets.
Our course of discrete mathematics consists of such
main parts:





 Graphs
 Basics of counting
 Types of algorithms and types
of problems
Set theory
Propositional logic
Predicate logic
Algebra of 2-valued elements
Boolean algebra
Discrete mathematics and computers.
Most methods of CDM are extremely useful for design and analysis of
computer software and hardware. These methods are the tools for improving
efficiency of existing software and hardware. On the other hand, modern
powerful computers give us hope of solving problems that were out of reach in
previous decades. It stimulates CDM to develop new methods for solving
problems.
2
Topic 1: Basic concepts of the set theory
The notion (concept) of a set is one of the initial notions of mathematics that cannot be
defined quite formally. We can talk about a set of triangle edges (sides), about the set of
Ukrainian words, about the set of natural numbers and so on. Synonyms of the term “set” are:
a collection, a group, a class, etc. Approximately, one may say that a set is a collection of
objects or ideas called elements of the set.
From the examples above, we can see that the notion of a set may be applied to collections of
different nature:
 A set of triangle sides (It contains a small, finite number of elements).
 A set of Ukrainian words (It contains a very large number, because nobody knows
how many Ukrainian words there exist. But, for sure, this number is less than, say,
one billion. So, this set is finite).
 A set of natural numbers (It contains an infinite amount of elements, because this set
can’t be listed even theoretically. But we can easily decide if a given number is a
natural one or not).
Sets are usually denoted by Latin capital letters and their elements – by corresponding small
letters. A = {a, b, c, d}, B = {b, d}, C = {a, b, c, d}.
Note: {a, b} = {b, a} - same unordered set, curly braces, sequence of elements not important.
(a, b) ≠ (b, a) – two different ordered sets, round brackets, sequence is important.
{a, b, b, c} ={a, b, c} – a set does not have duplicate elements if not stated otherwise.
If set B contains only elements that belong to set A then B is said to be a subset of A:
B  A (proper subset, not all elements of A are in B), C A (C may have all elements of
A or may not).
For any set A the following is true: ∅ ⊆ A, A⊆A.
Two sets A and B are equal (denoted by A = B) if they consist of the same elements. This is
equivalent to the condition that, for arbitrary x, if it is contained in A then it follows that x is
in set B as well, and vice versa. So:
For any xA (it follows) xB and for any yB  yA then A = B.
The number of elements in a set A is called the cardinality of A and is denoted │A│. If A =
{a, b, c, d} then │A│= 4.
Set P(A) is a power set of A. P(A) contains all possible subsets of set A.
Ex.: A = {a, b}. P(A) = {, {a}, {b}, {a, b}}. Elements of a power set are sets themselves.
There is a difference between “a” and “{a}”:
- a  A (A is an element of A).
- {a}  A; {a}  A, {a}  P(A). ({a} is not an element of A, it is a subset of A, and it
is an element of P(A).)
3
Example. Let the set of Ivory Coast population be {Alex, Bob, … , Sam, … }. Suppose Sam
is the only member of Ivory Cost team for some Winter Olympic games. In such a case the
set of national teams at the games = {{US team}, {Russia team}, {Canada team}, …, {Sam},
…}. Now Sam is treated as a team. He has a team status, like the status of the US Olympic
team of about 600 members.
Definition: sets A and B (finite or infinite) are called equivalent (A ≈ B) if they have the
same cardinality.
Ex.1:
A = {1, 7, 10, 15}, B = {a, b, c, d}. │A│= │B│⟹ A ≈ B.
Ex.2: C = {x│ x =𝑖 3 , i ∈ N}, D = {x│ x = i -3, i∈ N}. Note: N = {1, 2, 3 …} is the set of
natural numbers.
│C│= │N│, │D│ = │N│ ⟹ │C│ = │D│⟹ C ≈ D.
H/t: Prove that the number │P (A) │ of all subsets of a finite set A equals to 2│A│ .
Methods of set specification
a. Enumeration of set elements (listing element).
Ex.: D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} – set of decimal digits.
A = {a, b, c, d}.
b. Set comprehension (understanding/ describing the feature that unites elements in one
set).
C = {x│ x =𝑖 3 , i ∈ N} – set of cubes of natural numbers.
c. A set can be specified by means of set operations applied to initially available sets.
Ex.: 𝐴1 = {a, b}, 𝐵1 = {b, c, d}, 𝐷1 =𝐴1 ∪ 𝐵1; now 𝐷1 = {a, b, c, d}.
Operations on sets
It is convenient to illustrate operations on sets by means of Venn diagrams and Euler
diagrams. Such diagrams are made of Euler circles.
Note:
Euler diagrams only have the intersection combinations that actually exist in the real
world. Venn diagrams represent every hypothetically possible logical relation between
categories. Venn diagrams, by definition, have to display every possible intersection
combination, which creates some interesting layout issues.
4
Ex.: A = {a, b, c, d}, B = {a, b}, C = {c}. Euler diagram is made of 3 Euler circles.
A
C
B
Venn diagrams reflect the standard general relations between n sets.
It is also called Venn symbol. It has 2𝑛 domains, where n is the number of sets.
n=2
n=3
A
A B
C
B
and so on.
5
The union of sets A and B is a set C that contains only elements belonging to A or B. It is
denoted so:
C =𝐴 ∪ 𝐵. 𝐴 ∪ 𝐵 = {x│ x ∈ A or x ∈ B or both}.
It is pronounced “A union B” or “Union of A and B”.
Ex.:A = {a, b}, B = {b, c, d}, C=𝐴 ∪ 𝐵; now 𝐶 = {a, b, c, d}.
Union of two sets is represented by such a diagram:
A B
Denoted C = A ∪ B
The union itself consists of the domains that have any shading.
Set intersection
Intersection of sets A and B is a set C which contains elements that belong both to A and B.
Intersection is represented by such a diagram:
A B
Denoted: C = A∩B.
Intersection consists of the domain that has a darker shading.
C = A ∩ B = {x │ x ∈ A and x ∈ B}.
Ex.: A = {0, 1, 2, 3}, B = {0, 2, 4, 6}, C = {0, 2}.
Using the notion of the universal set one can define the operation of the compliment of a set.
Definition: A universal set contains all elements that are of interest in a given problem.
6
Set compliment
A compliment of a set A is a set B that contains all the elements of the universal set that does
not belong to A.
Compliment is represented by such a diagram:
Denoted Ā
A
Compliment consists of the domain without shading.
Ā = {x │x ∈ U and x∉ A}.
Ex.: U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {0, 2, 4, 6, 8}, Ā = {1, 3, 5, 7, 9}, where universal
set is a set of decimal digits, A is a set of even digits and Ā consists of odd digits.
Set difference
A difference of sets A and B is a set C that consists of the elements that belong to A and
don’t belong to B.
Difference is represented by such diagram:
A B
Denoted: A - B
Difference is represented by the domain with light shading.
A – B = {x │ x ∈ A and x∉ B}.
Ex.: A = {0, 1, 2, 3, 4}, B = 0, 2, 4, 6}, A - B = {1, 3}, B - A = {6}.
Properties of difference of two sets:
B - A≠A - B
B - A = Ā∩ B
The difference of two sets can be expressed by means of compliment and intersection of sets.
7
H/t: Draw a Venn diagram for sets:
A
(A∪B) ∩ B
(A - B) ∪(A ∩ B)
C
B
U
H/t: Given a Venn diagram.
Write a formula for each
of the domains.
Subsets of the universal set together with operations for sets constitute a special type of
algebra. It’s called set algebra.
Formulas of the set algebra transform sets into other sets.
Example of a formula of the set algebra:
(Ā∪ B) ∪ C.
It specifies a new set, consisting of elements of sets A, B and C.
Priority of operations: ¯, ∩, ∪, -.
8
Finite and infinite sets
Many sets discussed in mathematics are infinite. DM is interested only in countable sets (sets
where each element can get a unique counting number, that is elements are counted).
Ex.: A = {x | x=𝑖 2 , i ∈N} = {1, 4, 9, 16, 25…}. Note: N is the set of all natural numbers.
A is countable, so it is a discrete set. To show that A is countable, we establish a one-to-one
correspondence between the elements of A and N.
The doubly directed arrows represent the correspondence
between 𝒊 ∈ N and x ∈ A. It is the rule how to find x for a
given 𝒊 and vice versa.
Note: Not all infinite sets have the same cardinality. The set
of real numbers is not countable. It was proven that the set of
all real numbers from 0 to 1 contains infinitely more elements
than the whole set of natural numbers.
9
Vocabulary
10
Topic: “Set algebra”
commutative law – коммутативный закон
associative law – ассоциативный закон
distributive law – дистрибутивный закон
to be subject to – подчиняться
elimination law – закон элиминации
contradiction – противоречие
excluded middle – исключенное третье
Topic: “Relations”
Alink – связь
Cartesian product – декартово произведение
sequence – последовательность
Cartesian power of a set – декартова степень множества
notation – обозначение
triple – тройка (чего-либо)
arrangement – расположение
to satisfy – удовлетворять
matrix / matrices – матрица / матрицы
rules of precedence of operations – правила старшинства операций
Topic: “Operations on sets”
set union (union of sets) – объединение множеств
Venn diagram – диаграмма Венна
circle – окружность, круг
circumference – окружность / длина окружности
relation – отношение
relationship – взаимоотношение, связь
domain – область
hatch – штриховка
intersection – пересечение (множеств)
compliment (of a set) – дополнение
difference – разность
property – свойство
priority – приоритет, старшинство
11
Topic: “Basic concepts of the theory of sets”
a set – множество
to define – определить
a triangle – треугольник
an element of a set – элемент множества
finite – конечный
infinite – бесконечный
to denote – обозначать
negation – отрицание
compliment – дополнение
curly braces – фигурные скобки
a duplicate – дубликат
empty set – пустое множество
to contain – содержать
relevant – имеющий отношение, относящийся
universal set – универсальное множество
sub set – подмножество
proper sub set – собственное подмножество
to be equal – быть равным
arbitrary – произвольный
to imply – подразумевать, иметь в качестве следствия
cardinality – мощность множества
specify – задавать, уточнять
enumeration – перечисление
enumerate – перечислять
comprehension–понимание
12
set comprehension – задание множества описанием
× - Cartesian product
⊆- set inclusion
13