Intro & Applications of Discrete Math
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The document provides an overview of discrete mathematics and its applications. It begins by defining discrete mathematics as the study of mathematical structures that are discrete rather than continuous. Some key points made include: - Discrete mathematics deals with objects that can only assume distinct, separated values. Fields like combinatorics, graph theory, and computation theory are considered parts of discrete mathematics. - Research in discrete mathematics increased in the latter half of the 20th century due to the development of digital computers which operate using discrete bits. - The document then gives several examples of applications of discrete mathematics, such as in computer science, networking, cryptography, logistics, and scheduling problems. - Discrete mathematics is widely used in fields like
Intro & Applications of Discrete Math
- 1. SUBMITTED BY JUNAID SHIEKH GHULAM MUSTAFA MUBEEN AFTAB BILAL GOHAR (15074156-026) (15074156-100) (15074156-006) (15074156-028) SUBMITTED TO Mr. SAQIB UL ZAMAN 2 Intro & Applications of Discrete Math
- 2. Introduction to Discrete Math
- 3. What is discrete Structures? 3 • Discrete mathematics is the part of mathematics devoted to the study of discrete objects (Kenneth H. Rosen, 6th edition). • Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous (Wikipedia).
- 4. What is discrete Structures? 4 Discrete mathematics is the branch of mathematics dealing with objects that can assume only distinct, separated values. The study of how discrete objects combine with one another and the probabilities of various outcomes is known as combinatory. Other fields of mathematics that are considered to be part of discrete mathematics include graph theory and the theory of computation The study of topics in discrete mathematics usually includes the study of algorithms, their implementations, and efficiencies. Discrete mathematics is the mathematical language of computer science, and as such, its importance has increased dramatically in recent decades. Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in discrete steps and store data in discrete bits.
- 5. DISCREATE VS CONTINUES 5 • Examples of discrete Data - Number of boys in the class. - Number of candies in a packet. - Number of suitcases lost by an airline. • Examples of continuous Data - Height of a person. - Time in a race. - Distance traveled by a car.
- 6. Application of Discrete Math in real life 1
- 7. DISCRETE MATHINCOMPUTERS Computers run software and store files. The software and files are both stored as huge string of 1s and 0s Binary math is discrete mathematics. All computer data is represented using binary. Binary digits can be grouped together into bytes. 7
- 8. DISCRETE MATH IN RAILWAYPLANNING Railway planning uses discrete math deciding how to expand train rail lines, train timetable scheduling, and scheduling crews and equipment for train trips use both graph theory and linear algebra. 8
- 9. DISCRETE MATH IN COMPUTER GRAPHICS Computer graphics (such as in video games) use linear algebra in order to transform (move, scale, change perspective) objects. That's true for both applications like game development, and for operating system XY rotation matrix: 9
- 10. DISCRETE MATH INCRYPTOGRAPHY Cryptography is a method of storing and transmitting data in a particular form so that only those for whom it is intended can read and process it. Cryptography provide secure any data or passwords in encryption methods. computers send information in discrete -- or separate and distinct -- bits. Number theory, one important part of discrete math, allows cryptographers to create and break numerical passwords. Example :
- 11. DISCRETE MATH IN GOOGLE MAPS Google Maps uses discrete mathematics to determine fastest driving routes and times. There is a simpler version that works with small maps and technicalities involved in adapting to large maps. Google maps What is the shortest route from point A to point B? Any social networking site will have to use graph theory. 1 1
- 12. IMAGEPROCESSING Image processing is a method to convert an image into digital form and perform some operations on it, in order to get an enhanced image or to extract some useful information from it. It convert image as two dimensional signals. Digital image processing uses discrete mathematics to merge images or apply filters. 1 2
- 13. NETWORKS People are connected to each other by social media ("following" on Twitter, "friending" on Facebook, etc.). The US highway system connects with the roads. 1 3 Networks are, at base, discrete structures. The routers that run the internet are connected by long cables.
- 14. VOTINGSYSTEMS 1 4 There are different methods of voting -- not just the common cast-a-ballot-for- exactly-one-candidate method. The study of possible voting methods and how well their outcomes reflect the intent of the voters uses discrete mathematics.
- 15. DESIGNINGPASSWORD CRITERIA 1 5 Is the space of passwords chosen large enough that a hacker can't break into accounts just by trying all the possibilities. Discrete Math help decide how much possibilities are present to crack the password, so that we can make it more secure.
- 16. DISCRETE MATHIN LOGISTICS 1 6 Logistics is the study of organizing the flow of information, goods and services. Without discrete mathematics, logistics would not exist. This is because logistics makes heavy use of graphs and graph theory, a sub-field of discrete math. Graph theory allows complex logistical problems to simplify into graphs consisting of nodes and lines. A mathematician can analyze these graphs according to the methods of graph theory to determine the best routes for shipping or solving other logistical problems.
- 17. COMPUTERALGORITHMS 1 7 Algorithms are the rules by which a computer operates. These rules are created through the laws of discrete mathematics. 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. Because of discrete mathematical applications in algorithms, today’s computers run faster than ever before.
- 18. RELATIONAL DATABASES 1 8 Relational databases play a part in almost every organization that must keep track of employees, clients or resources. A relational database connects the traits of a certain piece of information. For example, in a database containing client information, the relational aspect of this database allows the computer system to know how to link the client’s name, address, phone number and other pertinent information. This is all done through the discrete math concept of sets. Sets allow information to be grouped and put in order. Since each piece of information and each trait belonging to that piece of information is discrete, the organization of such information in a database requires discrete mathematical methods.
- 19. WEB SEARCHES. 1 9 Doing web searches in multiple languages at once, and returning a summary, uses linear algebra. Web search involves Various Algorithms, which are called “Web Crawlers”. There job is to search every online page on the web. They do that through Graph Theory. They find the shortest path to the required page to increase speed.
- 20. SCHEDULINGPROBLEMS 2 0 Like deciding which nurses should work which shifts. Which airline pilots should be flying which routes. Scheduling rooms for an event Deciding timeslots for committee meetings Which chemicals can be stored in which parts of a warehouse All are solved either using graph coloring or using combinatorial optimization, both parts of discrete mathematics. One example is scheduling games for a professional sports league.
- 21. CELLPHONE COMMUNICATIONS 2 1 Cell phone communications: Making efficient use of the broadcast spectrum for mobile phones uses linear algebra and information theory. Assigning frequencies so that there is no interference with nearby phones can use graph theory or can use discrete optimization.
- 22. ARCHAEOLOGY 2 2 It uses discrete math to construct 3D images from scans of archaeological sites. Archaeologists and engineers are using lasers to scan many ancient structures to create three-dimensional images. Areas of mathematics such as vector analysis and linear algebra help convert the billions of measurements from the laser beams into coordinates and then align readings from repeated scans to achieve images that are accurate to within a millimeter.
- 23. MODELINGTRAFFIC 2 3 Determining the effect of regulation on network traffic flow---whether that's cars on roads or packets across the internet---is a matter of game theory and graph theory together. Optimizing traffic-light cycles uses both discrete and continuous mathematics.
- 24. POWER GRIDS 2 4 Graph theory is used in finding the most vulnerable aspects of an electric grid. Graph theory and linear algebra are used in power grid simulations. Differential equations, graph theory, and linear algebra help with the grid simulations that allow operators to monitor their systems and calculate system requirements. These simulations, involving millions of variables, can represent the grid as immediately as a few microseconds in the future, which allows operators to both sustain smooth performance and recover from any disruptions.
- 25. DATAMINING 2 5 Data mining is process of perceiving required information from huge data with the help of various methods. Mostly the data we deal with in data science can be shaped as graphs. e.g. in social networks every person in the network could be supposed as a vertex and any connection between them is supposed as an edge.
- 26. WEB DESIGNING 2 6 Website designing can be modeled as a graphs. Where the web pages are entitled by vertices and the hyper links between them are entitled by edges in the graph. This concept is called as web graph.
- 27. Researchandcorporateapplications that use discrete mathematics 2 7 Design of radar and sonar systems uses graph theory Robot arms are a type of linkage, the study of which is part of discrete geometry. Power grids: Graph theory is used in finding the most vulnerable aspects of an electric grid. Discrete math is used in choosing the most on-time route for a given train trip in the UK. Graph theory is used in cybersecurity to identify hacked or criminal servers and generally for network security. Graph theory is used in neuroscience to study brain network organization (see abstract here) and understand neuropathology for nervous system disorders.
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