One of the main aims of logic is to provide rules by whichone can validate whether any particular argument or reasoning is correct orincorrect.

Any collection of rules or any theory needs a language in whichthese rules or theory can be stated. From our daily experience we can say that natural languagesare not accurate as they can have different meaning. They are ambiguous and notsuitable for these purposes. Therefore we develop a formal language called theobject language.

In this language we use a well-defined object followed by adefinite statement regarding the same object. When we use mathematicalexpressions to denote the logical statements , we call this DiscreteMathematics , also commonly paired with Graph Theory.Discrete Mathematics is gaining popularity these daysbecause if it’s popularity and usage in computer science. Complex logic andcalculations can be depicted in the form of simple statements. It is used indaily life in the following ways :- 1.

)   The tasks running on computer use one or anotherform of discrete maths . The computer functions in a specific way depending onthe decisions made by the user. For example:Discrete Mathematics isvery closely connected with Computer Science.  Theoretical Computer      Science, the foundation of our field isoften considered a subfield of discrete mathematics.

  Computer Science isbuilt upon logic, and numerous, if not most, areas of discrete maths utilizedin the field.2.)   Discretemathematics describe processes that consist of a sequence of individual steps.Many ways of producing rankings use both discrete maths andgraph theory. Specific examples include rankingrelevance of search results using Google, ranking teamsfor tournaments or chicken pecking orders,and ranking sportsteam performances or restaurant preferences that include apparent paradoxen. 3.)   All of us write codes oncomputer on some platform with built in languages like C, Python,  Java etc.

but before writing the codes itselfwe prefer writing the algorithms, which involves basic logic for the code usingdiscrete maths. A computerprogrammer uses discrete math to design efficient algorithms. This designincludes applying discrete math to determine the number of steps an algorithmneeds to complete, which implies the speed of the algorithm.  Algorithms are the rules by which a computeroperates. These rules are created through the laws of discrete mathematics.Because of discrete mathematical applications in algorithms, today’s computersrun faster than ever before. Example of analgorithm:  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=0for j=0 to j=n-1               p = p +  return p {p is the value of ab}   We can clearlysee the application of logic and Discrete maths in the above algorithm.  4.

)   The field ofcryptography  is based entirely on discrete mathematics. Cryptography isthe study of how to create security structures and passwords for computers andother electronic systems. One of themost important part of discrete mathematics is Number theory which allowscryptographers to create and break numerical passwords.Shown belowis an example of Discrete Mathematics in encryption:   5.

)   Discretemathematics is being used in a really new way in the UK. Discretemath is used in choosing the moston-time route for a given train trip. The software is under development anduses discrete math to calculate the most time efficient route for a passenger.

 Each change oftrain by a passenger at a station is like an obstacle because of possibledelays, spreads out the arrival time of the passenger at the next station onthe route. For every part of the journey the kernel for each station is appliedin succession, giving the distribution of arrival time at the finaldestination.  Working of the system:-Each station hasa 60 x 60 matrix for a particular time of day.

It is 60 on one side because themaximum delay considered is an hour. The other side is 60 because that hour isdivided up into discrete one minute intervals, the nearest value provided bythe train timetables.The matrix isfitted with the probability that if you arrive at the station at minute i, youdepart at minute j. This is based on timetable information and the delayprofile information obtained from the website data grab. The matrices for eachstation are in turn applied to a column vector.

The column vector contains theprobability distribution of your arrival time at the next station with eachvalue showing the probability of being 0, 1,2, 3 minutes late etc. The totalcolumn vector sums to one. Before you depart, the first value in the columnvector is 1 and the rest are zeros – a delta function. This is because youhaven’t had chance to be subjected to delays yet.By applying yourstarting station’s matrix to this column vector, a new one is generatedcontaining the probability distribution of your arrival time at the nextstation.

The matrix forthat station is then applied to the new column vector, and so on until youreach your destination. The final, resultant column vector provides thedistribution of your probable arrival times. This can then be compared with thefinal column vector for other routes and the optimum route selected.   A railwaycontrol office using Mathematics and Graphs to analyse patterns.  6.)   Graphs arenothing but connected nodes(vertex).

So any network related, routing, findingrelation, path etc related real life applications use graphs. Aircraftscheduling: Assuming that there are k aircrafts and they have to be assigned nflights. The ith flight should be during the time interval (ai, bi). If twoflights 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 graphcorrespond to the flights. Two vertices will be connected, if the correspondingtime intervals overlap. Therefore, the graph is an interval graph that can becolored optimally in polynomial time.

 Belowis an example of the mathematical and graphical data used to check theoverlapping of various flights in a unanimous flying pattern so as to neglectcasualities.7.)   If you’ve ever used Google, you’re looking at the world’smost (financially) valuable graph theory application. At the heart of theirsearch engine technology is an algorithm called PageRank, which uses numerousgraph theory concepts — including cliques and a lot of connectivity information— to determine how important a given web page is.

It does this, in essence, bystarting with a rough notion of each page’s importance and then repeatedlyrefining its estimates by ‘flowing’ importance values from page to page.        THANK YOU

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