Lection 1.pptx

Lection 1
Topic 1: The basic concepts of
Discrete mathematics. Set Theory
Topic 2: Relations

The purpose of the course «Discrete
mathematics»
The purpose of the course is the formation of a
system of theoretical knowledge and practical
students’ skills with fundamentals of mathematical
apparatus for quantitative analysis of various
discrete masses phenomena that are characteristic
of economic objects; development of practical
abilities of students on the use of mathematical
language, the construction of mathematical models
and proofs, the implementation of mathematical
transformations for solving еconomic problems.

Basic literature
1. R. M. Trokhimchuk, M. S. Ni`ki`tchenko Diskretna matematika
u prikladakh i` zadachakh. - Kiyiv: Kiyiv. nacz. un-t i`m. Tarasa
Shevchenka, 2017. - 248 s.
2. Theoretical Computer Science and Discrete Mathematics : First
International Conference, ICTCSDM 2016, Krishnankoil, India,
December 19-21, 2016, Revised Selected Papers / edited by S.
Arumugam, Jay Bagga, Lowell W. Beineke, B.S. Panda. — 1st ed.
2017. — Cham : Springer International Publishing, 2017. — XIII, 458 p.
3. Guide to Discrete Mathematics: An Accessible Introduction to the
History, Theory, Logic and Applications / by Gerard O'Regan. — 1st ed.
2016. — Cham : Springer International Publishing, 2016.

Additional literature
1. Ni`kolayeva K.V., Kojbi`chuk V.V. Diskretnij anali`z :
navchal`nij posi`bnik. Sumi: UABS NBU, 2006. – 100 s.
2. Ni`kolayeva K.V., Kojbi`chuk V.V. Diskretnij anali`z : Grafi
ta yikh zastosuvannya v ekonomi`czi` Sumi: UABS NBU,
2007. – 84 s.
3. Susanna S. Epp Discrete mathematics with applications.
Fourth edition. 2011, Publisher, Richard Stratton, 685 p.
4. Algorithms and Discrete Applied Mathematics : Third
International Conference, CALDAM 2017, Sancoale, Goa,
India, February 16-18, 2017, Proceedings / edited by Daya
Gaur, N.S. Narayanaswamy. — 1st ed. 2017. — Cham :
Springer International Publishing, 2017. — XIX, 372 p.

Internet Resourses
1. L. Lovasz J., Pelikan K., Vesztergombi Discrete
Mathenlatics Elementary and Beyond. URL:
https://link.springer.com/content/pdf/10.1007%
2Fb97469.pdf .
2. Oscar Levin Discrete Mathematics. An Open
Introduction. URL :
http://discrete.openmathbooks.org/pdfs/dmoi-
tablet.pdf

Plan
1. Sets and Elements. Subsets. Laws of the algebra of sets.
2. Relations.
2.1. Product Sets.
2.2. Binary Relations.
2.3. Pictorial Representatives of Relations.
2.4. Inverse Relation.
2.5. Types of Relations.
2.6. Functional Relations.
2.7. One-to-one, onto, and Invertible Functions.
2.8. Ordered Sets.
2.9. Suplementary Problems.

General information about discrete
mathematics
• Discrete mathematics is the study of mathematical structures that
are fundamentally discrete rather than continuous. In contrast
to real numbers that have the property of varying "smoothly", the
objects studied in discrete mathematics – such as integers, graphs,
and statements in logic – do not vary smoothly in this way, but have
distinct, separated values.
• Discrete mathematics therefore excludes topics in "continuous
mathematics" such as calculus or Euclidean geometry. Discrete
objects can often be enumerated by integers. More formally,
discrete mathematics has been characterized as the branch of
mathematics dealing with countable sets (finite sets or sets with
the same cardinality as the natural numbers). However, there is no
exact definition of the term "discrete mathematics." Indeed,
discrete mathematics is described less by what is included than by
what is excluded: continuously varying quantities and related
notions.

The originators of the basic concepts of Discrete Mathematics, the
mathematics of finite structures, were the Hindus, who knew the formulae for
the number of permutations of a set of n elements, and for the number of
subsets of cardinality k in a set of n elements already in the sixth century. The
beginning of Combinatorics as we know it today started with the work of
Pascal and De Moivre in the 17th century, and continued in the 18th century
with the seminal ideas of Euler in Graph Theory, with his work on partitions
and their enumeration, and with his interest in latin squares. These old results
are among the roots of the study of formal methods of enumeration, the
development of configurations and designs, and the extensive work on Graph
Theory in the last two centuries. The tight connection between Discrete
Mathematics and Theoretical Computer Science, and the rapid development
of the latter in recent years, led to an increased interest in Combinatorial
techniques and to an impressive development of the subject. It also stimulated
the study and development of algorithmic combinatorics and combinatorial
optimization.
General information about discrete mathematics

Concepts and questions of Discrete Mathematics appear
naturally in many branches of mathematics, and the area
has found applications in other disciplines as well. These
include applications in Information Theory and Electrical
Engineering, in Statistical Physics, in Chemistry and
Molecular Biology, and, of course, in Computer Science.
Combinatorial topics such as Ramsey Theory,
Combinatorial Set Theory, Matroid Theory, Extremal
Graph Theory, Combinatorial Geometry and Discrepancy
Theory are related to a large part of the mathematical
and scientific world, and these topics have already found
numerous applications in other fields.

It seems safe to predict that in the future Discrete
Mathematics will be continue to incorporate
methods from other mathematical areas. However,
such methods usually provide non-constructive
proof techniques, and the conversion of these to
algorithmic ones may well be one of the main
future challenges of the area (involving cooperation
with theoretical computer scientists). Another
interesting recent development is the increased
appearance of computer-aided proofs in
Combinatorics, starting with the proof of the Four
Color Theorem.

The Importance of
Discrete Mathematics
Discrete mathematics is the branch of mathematics
dealing with objects that can assume only distinct,
separated values. Discrete means individual, separate,
distinguishable implying discontinuous or not continuous,
so integers are discrete in this sense even though they are
countable in the sense that you can use them to count.
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 (and which includes, for example, calculus).
Whereas discrete objects can often be characterized by
integers, continuous objects require real numbers.

Discrete Mathematics is the backbone
of Computer Science
Discrete mathematics has become popular in recent
decades because of its applications to computer science.
Discrete mathematics is the mathematical language of
computer science. Concepts and notations from discrete
mathematics are useful in studying and describing objects
and problems in all branches of computer science, such
as computer algorithms, programming
languages, cryptography, automated theorem proving,
and software development. Conversely,
computer implementations are tremendously significant
in applying ideas from discrete mathematics to real-world
applications, such as in operations research.

The set of objects studied in discrete mathematics
can be finite or infinite. In real-world applications,
the set of objects of interest are mainly finite, the
study of which is often called finite mathematics. In
some mathematics curricula, the term “finite
mathematics” refers to courses that cover discrete
mathematical concepts for business, while “discrete
mathematics” courses emphasize discrete
mathematical concepts for computer science
majors.

Discrete math plays the significant role in big data analytics
The Big Data era poses a critically difficult challenge and striking
development opportunities: how to efficiently turn massively
large data into valuable information and meaningful knowledge.
Discrete mathematics produces a significant collection of
powerful methods, including mathematical tools for
understanding and managing very high-dimensional data,
inference systems for drawing sound conclusions from large and
noisy data sets, and algorithms for scaling computations up to
very large sizes. Discrete mathematics is the mathematical
language of data science, and as such, its importance has
increased dramatically in recent decades.

IN SUMMARY
Discrete mathematics is an exciting and appropriate
vehicle for working toward and achieving the goal
of educating informed citizens who are better able
to function in our increasingly technological society;
have better reasoning power and problem-solving
skills; are aware of the importance of mathematics
in our society; and are prepared for future careers
which will require new and more sophisticated
analytical and technical tools. It is an excellent tool
for improving reasoning and problem-solving
abilities.

Laws of the algebra of sets (Table 1.1.1)

Power set
In mathematics, the power set (or powerset) of
any set S is the set of all subsets of S, including
the empty set and S itself, variously denoted
as P(S), 𝒫(S), ℘(S) (using the "Weierstrass
p"), P(S), ℙ(S), or, identifying the powerset
of S with the set of all functions from S to a
given set of two elements, 2S

Example
If S is the set {x, y, z}, then the subsets of S are:
• {} (also denoted ∅, the empty set or the null set)
• {x}
• {y}
• {z}
• {x, y}
• {x, z}
• {y, z}
• {x, y, z}
and hence the power set of S is {{}, {x}, {y}, {z}, {x, y},
{x, z}, {y, z}, {x, y, z}}.

Representing subsets as functions
• In set theory, XY is the set of all functions from Y to X. As "2" can be
defined as {0,1} (see natural number), 2S (i.e., {0,1}S) is the set of
all functions from S to {0,1}. By identifying a function in 2S with the
corresponding preimage of 1, we see that there is
a bijection between 2S and P(S), where each function is the characteristic
function of the subset in P(S) with which it is identified. Hence 2S and P(S)
could be considered identical set-theoretically. (Thus there are two
distinct notational motivations for denoting the power set by 2S: the fact
that this function-representation of subsets makes it a special case of
the XY notation and the property, mentioned above, that |2S| = 2|S|.)
• This notion can be applied to the example above in which S = {x, y, z} to
see the isomorphism with the binary numbers from 0 to 2n −
1 with n being the number of elements in the set. In S, a "1" in the
position corresponding to the location in the enumerated set { (x, 0), (y, 1),
(z, 2) } indicates the presence of the element. So {x, y} = 011(2).

For the whole power set of S we get:
Subset
Sequence
of digits
Binary
interpretation
Decimal
equivalent
{ } 0, 0, 0 000(2) 0(10)
{ x } 0, 0, 1 001(2) 1(10)
{ y } 0, 1, 0 010(2) 2(10)
{ x, y } 0, 1, 1 011(2) 3(10)
{ z } 1, 0, 0 100(2) 4(10)
{ x, z } 1, 0, 1 101(2) 5(10)
{ y, z } 1, 1, 0 110(2) 6(10)
{ x, y, z } 1, 1, 1 111(2) 7(10)
Such bijective mapping of S to integers is arbitrary, so this representation of subsets of S is
not unique, but the sort order of the enumerated set does not change its cardinality.
However, such finite binary representation is only possible if S can be enumerated (this is
possible even if S has an infinite cardinality, such as the set of integers or rationals, but not
for example if S is the set of real numbers, in which we cannot enumerate all irrational
numbers to assign them a defined finite location in an ordered set containing all irrational
numbers).

Cardinality and Countability
We say the sets A and B have the same size
or cardinality if there is a bijection f:A→B . If this
is the case we write A≈B.
A function f:A→B is bijective (or f is a bijection)
if each b∈B has exactly one preimage. Since "at
least one'' + "at most one'' = "exactly one'', f is a
bijection if and only if it is both an injection and
a surjection. A bijection is also called a one-to-
one correspondence.

Example of bijection
If A={1,2,3,4} and B={r,s,t,u}, then
• f(1)=u
• f(2)=r
• f(3)=t
• f(4)=s
is a bijection

Examples of invertible functions
1) If f is the function
• f(1)=u f(2)=r f(3)=t f(4)=s
and
• g(r)=2 g(s)=4 g(t)=3 g(u)=1
then f and g are inverses.
• For example, f(g(r))=f(2)=r and g(f(3))=g(t)=3.
2) An inverse to x5 is 5
𝑥:
5
𝑥 5
= 𝑥
5
𝑥5 = 𝑥

Cardinality and Countability
If A and B are finite, then A≈B if and only
if A and B have the same number of elements.
This example shows that the definition of
"same size'' extends the usual meaning for finite
sets, something that we should require of any
reasonable definition.
We say a set A is countably infinite if N≈A,
that is, A has the same cardinality as the natural
numbers. We say A is countable if it is finite or
countably infinite.

The pigeonhole principle
The pigeonhole principle is a powerful tool used in
combinatorial math. But the idea is simple and can be explained by the
following peculiar problem.
Imagine that 3 pigeons need to be placed into 2 pigeonholes.
Can it be done? The answer is yes, but there is one catch. The catch is
that no matter how the pigeons are placed, one of the pigeonholes
must contain more than one pigeon.
The logic can be generalized for larger numbers. The
pigeonhole principle states that if more than n pigeons are placed
into n pigeonholes, some pigeonhole must contain more than one
pigeon. While the principle is evident, its implications are astounding.
The reason is that the principle proves the existence (or impossibility)
of a particular phenomenon.

The pigeonhole principle states that if n items are put
into m containers, with n>m, then at least one container must contain
more than one item. In layman's terms, if you have more "objects"
than you have "holes," at least one hole must have multiple objects in
it. A real-life example could be, "if you have three gloves, then you
have at least two right-hand gloves, or at least two left-hand gloves,"
because you have 3 objects, but only two categories to put them into
(right or left). This seemingly obvious statement, a type of counting
argument, can be used to demonstrate possibly unexpected results.
For example, if you know that the population of London is greater than
the maximum number of hairs that can be present on a human's head,
then the pigeonhole principle requires that there must be (at least)
two people in London who have the same number of hairs on their
heads.

Although the pigeonhole principle appears as early
as 1624 in a book attributed to Jean Leurechon, it is
commonly called Dirichlet's box principle or Dirichlet's
drawer principle after an 1834 treatment of the principle
by Peter Gustav Lejeune Dirichlet under the
name Schubfachprinzip ("drawer principle" or "shelf
principle").
The principle has several generalizations and can
be stated in various ways. In a more quantified version:
for natural numbers k and m, if n=km+1 objects are
distributed among m sets, then the pigeonhole principle
asserts that at least one of the sets will contain at
least k+1 objects.

Applications (examples) of the pigeonhole
principle
1) If you pick five cards from a standard deck of 52 cards, then at least
two will be of the same suit.
• Each of the five cards can belong to one of four suits. By the
pigeonhole principle, two or more must belong to the same suit.
2) If you have 10 black socks and 10 white socks, and you are picking
socks randomly, you will only need to pick three to find a matching
pair.
• The three socks can be one of two colors. By the pigeonhole
principle, at least two must be of the same color.
• Another way of seeing this is by thinking sock by sock. If the second
sock matches the first, then we are done. Otherwise, pick the third
sock. Now the first two socks already cover both color cases. The
third sock must be one of those and form a matching pair.

Applications (examples) of the pigeonhole
principle
3) If you pick five numbers from the integers 1 to 8, then two of
them must add up to nine.
• Every number can be paired with another to sum to nine. In
all, there are four such pairs: the numbers 1 and 8, 2 and 7, 3
and 6, and lastly 4 and 5.
• Each of the five numbers belongs to one of those four pairs.
By the pigeonhole principle, two of the numbers must be
from the same pair–which by construction sums to 9.

2.3. Pictorial Representatives of Relations

Types of Relations
Definition. A binary relation R on a set A is called symmetric if whenever aRb
then bRa, that is whenever a, b R then b, a R .
Definition. A binary relation R on a set A is called antisymmetric if whenever
aRb and bRa then a = b, that is, if a  b and aRb then bRa
Definition. A binary relation R on a set A is called transitve if whenever aRb
and bRc then aRc, that is, if whenever a, b, b, c R then a, c R.
Definition. A binary relation R on a set A is called complete if whenever a  A
and b  B then a = b, or a, b R. , or b, a R. .

Example
Determine whether or not each of the above
relations on A is:
reflexive; 2) symmetric; 3) transitive; 4)
antysymmetric.
Example 2.5.2.Consider the following five relations on the set
R  1,1,1,2, 1,3, 3,3, S  1,1,1,2, 2,1, 2, 2, 3,3

2.7. One-to-one, onto, and Invertible Functions

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