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One of the main aims of logic is to provide rules by which
one can validate whether any particular argument or reasoning is correct or
incorrect. Any collection of rules or any theory needs a language in which
these rules or theory can be stated.

From our daily experience we can say that natural languages
are not accurate as they can have different meaning. They are ambiguous and not
suitable for these purposes. Therefore we develop a formal language called the
object language. In this language we use a well-defined object followed by a
definite statement regarding the same object. When we use mathematical
expressions to denote the logical statements , we call this Discrete
Mathematics , also commonly paired with Graph Theory.

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Discrete Mathematics is gaining popularity these days
because if it’s popularity and usage in computer science. Complex logic and
calculations can be depicted in the form of simple statements. It is used in
daily life in the following ways :-

 

1.)   
The tasks running on computer use one or another
form of discrete maths . The computer functions in a specific way depending on
the decisions made by the user. For example:

Discrete Mathematics is
very closely connected with Computer Science.  Theoretical Computer      Science, the foundation of our field is
often considered a subfield of discrete mathematics.  Computer Science is
built upon logic, and numerous, if not most, areas of discrete maths utilized
in the field.

2.)   
Discrete
mathematics describe processes that consist of a sequence of individual steps.
Many ways of producing rankings use both discrete maths and
graph theory. Specific examples include ranking
relevance of search results using Google, ranking teams
for tournaments or chicken pecking orders,
and ranking sports
team performances or restaurant preferences that include apparent paradoxen.

 

3.)   
All of us write codes on
computer on some platform with built in languages like C, Python,  Java etc. but before writing the codes itself
we prefer writing the algorithms, which involves basic logic for the code using
discrete maths.

 

A computer
programmer uses discrete math to design efficient algorithms. This design
includes applying discrete math to determine the number of steps an algorithm
needs to complete, which implies the speed of the algorithm.  Algorithms are the rules by which a computer
operates. These rules are created through the laws of discrete mathematics.
Because of discrete mathematical applications in algorithms, today’s computers
run faster than ever before.

 

Example of an
algorithm:

 

 

procedure multiply(a ,
b:positive integers)

 {the binary expansions of a and b are ( ) and ( )
respectively}

for j=0 to j=n-1

               if  then  shifted j places

               else

{ }

p=0

for j=0 to j=n-1

               p = p +  

return p {p is the value of ab}

 

 

We can clearly
see the application of logic and Discrete maths in the above algorithm.

 

 

4.)   
The field of
cryptography  is based entirely on discrete mathematics. Cryptography is
the study of how to create security structures and passwords for computers and
other electronic systems.

One of the
most important part of discrete mathematics is Number theory which allows
cryptographers to create and break numerical passwords.

Shown below
is an example of Discrete Mathematics in encryption:

 

 

 

5.)   
Discrete
mathematics is being used in a really new way in the UK. Discrete
math is used in choosing the most
on-time route for a given train trip. The software is under development and
uses discrete math to calculate the most time efficient route for a passenger.

 

Each change of
train by a passenger at a station is like an obstacle because of possible
delays, spreads out the arrival time of the passenger at the next station on
the route. For every part of the journey the kernel for each station is applied
in succession, giving the distribution of arrival time at the final
destination.

 

 

Working of the system:-

Each station has
a 60 x 60 matrix for a particular time of day. It is 60 on one side because the
maximum delay considered is an hour. The other side is 60 because that hour is
divided up into discrete one minute intervals, the nearest value provided by
the train timetables.

The matrix is
fitted with the probability that if you arrive at the station at minute i, you
depart at minute j. This is based on timetable information and the delay
profile information obtained from the website data grab. The matrices for each
station are in turn applied to a column vector. The column vector contains the
probability distribution of your arrival time at the next station with each
value showing the probability of being 0, 1,2, 3 minutes late etc. The total
column vector sums to one. Before you depart, the first value in the column
vector is 1 and the rest are zeros – a delta function. This is because you
haven’t had chance to be subjected to delays yet.

By applying your
starting station’s matrix to this column vector, a new one is generated
containing the probability distribution of your arrival time at the next
station.

The matrix for
that station is then applied to the new column vector, and so on until you
reach your destination. The final, resultant column vector provides the
distribution of your probable arrival times. This can then be compared with the
final column vector for other routes and the optimum route selected.

 

 

 

A railway
control office using Mathematics and Graphs to analyse patterns.

 

 

6.)   
Graphs are
nothing but connected nodes(vertex). So any network related, routing, finding
relation, path etc related real life applications use graphs. Aircraft
scheduling: Assuming that there are k aircrafts and they have to be assigned n
flights. The ith flight should be during the time interval (ai, bi). If two
flights overlap, then the same aircraft cannot be assigned to both the flights.
This problem is modeled as a graph as follows. The vertices of the graph
correspond to the flights. Two vertices will be connected, if the corresponding
time intervals overlap. Therefore, the graph is an interval graph that can be
colored optimally in polynomial time.

 

7.)   
If you’ve ever used Google, you’re looking at the world’s
most (financially) valuable graph theory application. At the heart of their
search engine technology is an algorithm called PageRank, which uses numerous
graph theory concepts — including cliques and a lot of connectivity information
— to determine how important a given web page is. It does this, in essence, by
starting with a rough notion of each page’s importance and then repeatedly
refining its estimates by ‘flowing’ importance values from page to page.

 

 

 

 

 

 

 

 

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