Discrete mathematics is used in many real-life applications such as Google Maps, DNA sequencing, and modeling disease spread. Google Maps uses discrete structures like graphs and sets, as well as algorithms like Dijkstra's, to determine optimal routes. DNA sequencing relies on graph theory approaches like Hamiltonian paths and Euler paths to reconstruct DNA sequences from fragments. Disease spread is modeled using graph theory and discrete models to analyze transmission patterns and predict outbreaks.

1. Real Life Application of
Discrete Mathematics

2. Yash Bhattad
202251162
Rushikesh
Chaudhari
202251113
Manav Mehta
202252344
Team : Math GPT
Course Instructor : Swapnil Lokhande Sir

3. What do we really mean by
Discrete Mathematics ?
Discrete mathematics is a branch of
mathematics that deals with the
study of mathematical structures
that are discrete rather than
continuous, such as integers,
graphs, sets, and sequences.

4. Discrete Mathematics is
applicable in real life!
Whoa!

5. Discrete Mathematics is used in following topics:
1) Google Map/ Traffic Modeling
2) In computer programming
3) Computer Graphics
4) Image Processing
5) DNA Sequencing / Biometrics
6) Digital Electronics (Automatic Doors / Microprocessors / Fitness Trackers)
7) Game Theory
8) Railway Planning
9) Analog Clock
10) Voting Systems
11) Power Grids
12) Cyber Security/ Data encryption

6. COVID-19
01

7. Analysis of Spread
Graph theory and network analysis are
important tools for modeling the spread of
infectious diseases. Graph theory is the
study of graphs, which are mathematical
structures used to model relationships
between objects. Graphs can be used to
represent the spread of the virus by
mapping out the connections between
infected individuals.
The analysis of the graph can then be used
to identify patterns and trends in the
spread of the virus, which can be used to
develop strategies for controlling its
spread.

8. Predicting Spread
Discrete mathematics can also be used to
develop models for predicting the spread of the
virus. These models can be used to estimate the
rate of spread and the potential for a second
wave of infections. The models can also be used
to predict the impact of interventions, such as
social distancing, on the spread of the virus.
By using discrete mathematics to develop
models for predicting the spread of the virus,
public health officials can more effectively
manage the pandemic and develop
strategies for controlling its spread.

9. Contact Tracing
Algorithm
Discrete mathematics can be used to
model and analyze contact tracing
data to identify and track potential
contacts of an infected individual.
This can help to reduce the spread of
disease and identify high-risk areas.
Discrete mathematics can also be
used to develop algorithms for
contact tracing, such as those used in
mobile applications. These
algorithms can be used to identify
and alert individuals who may have
been exposed to a contagious
disease.

10. Contact Tracing Algorithm
Contact tracing algorithms may use a
variety of data sources to identify close
contacts, including GPS location data,
Bluetooth signals, and data from public
health agencies. Privacy concerns are
important, and contact tracing
algorithms aim to balance the need to
identify potential exposures with
protecting individual privacy.

11. Digital
Electronics
02

12. Introduction to Discrete
Math in Digital Electronics
It is used in the study of digital electronics,
which is the study of electronic circuits
that are composed of discrete
components. Digital electronics is used in
many aspects of modern technology,
including computers, cell phones, and
other electronic devices.
Digital electronics is based on the binary
system, which is a system of two values, 0
and 1. In this system, each digit is
represented by a single bit, which can be
either 0 or 1. This allows for the
representation of complex data in a simple
form.

13. Boolean Algebra
Boolean algebra is a fundamental
concept in digital electronics, and it is
used to manipulate and simplify logic
expressions used in digital circuit design.
Digital electronic circuits are built using
digital components like transistors,
diodes, and gates that operate based on
the principles of Boolean algebra.
The use of Boolean algebra in digital
electronics helps designers to ensure that
the circuits they create are accurate,
reliable, and efficient. It also allows for
easier testing and troubleshooting of
digital circuits.

14. Boolean algebra allows the
designer to manipulate these
binary signals and create more
complex logic functions using
logic gates such as AND, OR,
NOT, NAND, NOR, and XOR.
For example, the logical
expression "A AND B" can be
implemented using an AND
gate, which outputs a 1 only if
both A and B inputs are 1.

15. Boolean Algebra

16. Boolean Algebra

17. DNA
Sequencing
03

18. DNA SEQUENCING
The main goal of DNA sequencing is to decode the genetic information
contained within a DNA sequence, which is crucial for many fields of
research, including genetics, molecular biology, medicine, and forensic
science. Discrete mathematics plays a significant role in DNA sequencing,
which is the process of determining the precise order of nucleotides within
a DNA molecule.

19. Hamiltonian Path
Approach
Methods of approach of
DNA Sequencing based
upon Graph Theory
There are 2 different methods used for
Basic DNA Sequencing
Euler's Path
Approach

20. Hamiltonian Path Approach
First and foremost, A Hamiltonian path in a graph is a
path which visits each vertex in the graph exactly once.
In this method we will take all fragments as nodes of
the graph, and each node will connect if the ‘k-1’
rightmost nucleotide of the first note will overlap with
the ‘k-1’ leftmost nucleotide of the second node,
basically by connecting a suffix and a prefix that
overlap each other.
Let the Sample of DNA Reads be:

21. Here we have
a Hamiltonian
Cycle

22. Euler’s Path Approach
An Euler Path is a path that goes through every edge of
a graph exactly once.
In this approach we will use distinct prefix/suffix from
the reads as a node and each oligonucleotide becomes
an which has an initial endpoint ‘k-1’ from the
rightmost nucleotide of arc and its terminal end point
being ‘k-1’ of leftmost nucleotide.
Let's take the sample size of DNA reads as we have
taken for Hamiltonian Path Approach

23. Genome: ATGGCGTGCA
We obtain the same sequence of reads as before,
thus we will obtain the same sequenced genome
as before!

24. Google Maps
04

25. Google Maps uses discrete
mathematics application to
determine the shortest distance
between two locations( by Dijkstra's
algorithm explain in later slides)
Google Maps uses set theory to
categorize different locations and
points of interest.By categorizing
locations into sets, such as
restaurants, hotels, and gas stations,
Google Maps can provide users with
customized search results based on
their preferences.
Google Maps uses probability to
predict traffic patterns and estimate
arrival times.

26. Dijkstra's algorithm
Dijkstra's algorithm is a way of finding the
shortest path between two points on a
map,it is also use in network routing, GPS
navigation systems, and social network
analysis.
It works by treating the map as a
network of connected points, and
assigning each point a "distance" value
that represents how far it is from the
starting point.

27. 1. Create a graph with nodes and edges, where each edge has a
weight or cost.
2. a starting node and assign it a distance of 0. Assign a distance of
infinity to all other nodes in the graph.
3. Mark all nodes as unvisited and create an empty set to hold
visited nodes.
4. For the current node, calculate the tentative distance from the
starting node to each of its neighboring nodes. The tentative
distance is the sum of the current node's distance and the cost of
the edge between the current node and its neighbor.
This are the steps of Dijkstra’s algorithm :

28. 5. Compare the tentative distance to each neighbor's current
distance. If the tentative distance is less than the current distance, update
the neighbor's distance with the new tentative distance.
6. Mark the current node as visited and remove it from the unvisited set.
7. If the destination node has been visited or if the smallest tentative
distance among the unvisited nodes is infinity, stop the algorithm.
8. Select the unvisited node with the smallest tentative distance as the
next current node,and go back to step 4.
9. Backtrack from the destination node to the starting node to find the
shortest path
Let’s take an example

29. 1
1
3
4
5
4
4
2
2
3
3
0 ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Dist[ ] :
( dist , node )
(0,0)
0
2

30. 1
0
1
2
3
4
5
4
4
2
2
3
3
0 4 4 7 ∞ ∞
0 1 2 3 4 5
Dist[ ] :
( dist , node )
(7,3)
(4,2)
(4,1)
(0,0)
From 0 to 2 we have 4 unit so (4,2).
If we get the shorter path than the current path we update our
distance so infinity becomes 4 at 1st node same happen with
node at 2 and 3. previousThis cycle continue till all points gets
their shortest distance.

31. 1
3
4
4
2
2
3
3
0 4 4 7 5 8
0 1 2 3 4 5
Dist[ ] :
( dist , node )
(8,5)
(10,5)
(5,4)
(7,3)
(4,2)
(4,1)
(0,0)
0
1
2
4
5
So the shortest path between 0 to 5 is 0–2–4–5 which is 8 unit long.

32. Bibliography:
Discrete Mathematics in Life Science using probability
distribution:
math.berkeley.edu/~qchu/Notes/239.pdf
Explanation to Euler’s & Hamiltonian Path:
https://meenalpathak.wordpress.com/Euler'sTrail/Ha
miltonian'sPath
There are some other algorithm that may be better
than Dijkstra’s algorithm in some cases:
1) A* Algorithm :
https://en.wikipedia.org/wiki/A*_search_algorithm
2) Bellman-Ford Algorithm:
https://www.baeldung.com/cs/bellman-ford
3) Floyd-Warshall Algorithm :
https://brilliant.org/wiki/floyd-warshall-algorithm

33. Conclusion:
The Application of discrete mathematics is
widespread and can be found in various aspects of
our daily lives.Discrete mathematics also plays a
vital role in the development of artificial
intelligence and machine learning, which are
becoming increasingly important in our modern
world. Therefore, the importance of discrete
mathematics in real life cannot be overstated, and
its continued development will undoubtedly
contribute to the advancement of many fields and
improve our daily lives in numerous ways.

34. THANK YOU
THANK YOU