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.

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.

Below

is an example of the mathematical and graphical data used to check the

overlapping of various flights in a unanimous flying pattern so as to neglect

casualities.

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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