This document introduces discrete mathematics and its applications. It discusses how discrete mathematics underlies computer science and problem solving. Discrete structures are used to represent and manipulate digital data. The document outlines the main topics covered in a discrete mathematics course, including sets, relations, groups, graphs, trees, and discrete probability. It provides examples of how these concepts are applied to domains like social networks, software design, and routing algorithms.

1. APPLICATIONS OF DISCRETE
STRUCTURES

2. IntroductionCSCE 235, Spring 2010 2
WHY DISCRETE MATHEMATICS? (I)
Computers use discrete structures to
represent and manipulate data.
Computer Science is not Programming
Computer Science is not Software
Engineering
Edsger Dijkstra: “Computer Science is no
more about computers than Astronomy is
about telescopes.”
Computer Science is about problem
solving.

3. IntroductionCSCE 235, Spring 2010 3
WHY DISCRETE MATHEMATICS? (II)
 Mathematics is at the heart of problem
solving
 Defining a problem requires mathematical
rigor
 Use and analysis of models, data structures,
algorithms requires a solid foundation of
mathematics
 To justify why a particular way of solving a
problem is correct or efficient (i.e., better
than another way) requires analysis with a
well-defined mathematical model.

4. IntroductionCSCE 235, Spring 2010 4
APPLICATIONS(1)
 Discrete mathematics describes processes that
consist of a sequence of individual steps. This
contrasts with calculus, which describes
processes that change in a continuous fashion.
 Whereas the ideas of calculus were fundamental
to the science and technology of the industrial
revolution, the ideas of discrete mathematics
underlie the science and technology of the
computer age.

5. IntroductionCSCE 235, Spring 2010 5
APPLICATIONS(2)
 The main themes of a first course in discrete
mathematics are logic and proof, induction and
recursion, discrete structures, combinatorics and
discrete probability, algorithms and their analysis,
and applications and modeling.

6. IntroductionCSCE 235, Spring 2010 6
UNIT I SETS AND PROPOSITIONS
 This unit help students develop the ability to think
abstractly.
 This means learning to use logically valid forms of
argument and avoid common logical errors
 Set theory is the foundation of mathematics.

7. Unit II Relations and Functions
1 to many
1 to 1 many to many

8. IntroductionCSCE 235, Spring 2010 8
UNIT III GROUPS AND RINGS
 Problems in this field often arise (or follow naturally
from) a problem that is easily stated involving
counting, divisibility, or some other basic arithmetic
operation. While many of the problems are easily
stated, the techniques used to attack these
problems are some of the most difficult and
advanced in mathematics.

9. IntroductionCSCE 235, Spring 2010 9
UNIT IV GRAPH THEORY
 “EVERYTHING IS A GRAPH”
(labeled, directed, etc., ...)
graph theory can be used in modelling of:
Social networks
Communications networks
Information networks
Software design
Transportation networks
Biological networks

10. IntroductionCSCE 235, Spring 2010 10
UNIT V TREES
1. Manipulate hierarchical data.
2. Make information easy to search (see tree
traversal).
3. Manipulate sorted lists of data.
4. As a workflow for compositing digital images for
visual effects.
5. Router algorithms

11. IntroductionCSCE 235, Spring 2010 11
UNIT VI PERMUTATIONS, COMBINATIONS AND
DISCRETE PROBABILITY
 A combination is a selection of all or part of a set
of objects, without regard to the order in which they
were selected. This means that XYZ is considered
the same combination as ZYX.
 A permutation is an arrangement of all or part of a
set of objects, with regard to the order of the
arrangement. This means that XYZ is considered a
different permutation than ZYX.
 The probability of an event refers to the likelihood
that the event will occur