## Discrete a and b are ( ) and

Discrete
Mathematics is the study of mathematics limited to set of integers. Discrete
Mathematics is becoming the basis of many real world problems particularly in
computer science.

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 coding
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 of it’s increasing 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.)  Algorithms

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
math applied 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, nowadays 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 0

{ }

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.

2.)  Cryptography

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

Because of the amount of money and
the amount of confidential information involved, cryptographers must first have
a solid background in number theory to show they can provide secure passwords
and encryption methods.

Shown below is an example of
Discrete Mathematics in encryption:

3.) Computer Programmes

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 mathematics utilized in the field.

For Example:

p(x) denote “number x+4 is an even
integer”

~p(x) denote “number x+4 is not an
even integer”

q(x,y) to represent an open
statement that contains 2 variables.

With p(x) and q(x,y) as above,
universe still concern itself with integers only, make replacements for x,y we
get:

p(5) = (5+2) is an even integer

~p(7) = (7+2) is not an even
integer

q(4,2) = numbers 4,2,8 are even
integers

“For some x” and “For some x,y”
are said to quantify the open statement p(x) and q(x,y) respectively

·      For some x, p(x)

·      For some x,y  q(x,y)

4.) Computing
Rankings

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

5.) Train Delay

Discrete
mathematics is 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. On the other side it is 60 because the
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.)  Airplane Deviation

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 causalities and
deviation of flights:

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.

8.)   Relational Database

They
play a important part in almost every organization that keep track of it’s
employees, clients or resources. A relational database helps to join different
piece of information.

This is
all done with the concept of sets in discrete math. Sets allow the info to be
grouped and put together.

For
example:

Database
other info.

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Archit Goyal

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