Discrete mathematics for Computer Science
Discrete mathematics is used to provide good knowledge across every area of computer science. In computer science, the applications of discrete mathematics are very vast and described as follows:
The bits like one or zero is used to represent all data of the computer at a most fundamental level. Computers perform the calculations when they do some modification in these bits in accordance with Boolean algebra law, which is used to form all digital circuits. The graph is used to represent that digital circuit. The logical operators like and, or, and not are used to develop low-level programming languages. When software developers develop any project, they mostly prefer high-level languages. Sometimes they want to optimize their code by reducing the low-level operations, and sometimes, they also directly operate on bits. Programmers can also control the program flow by using Boolean logic. That means they can define certain conditions and then control which instructions will be executed.
Probability is used in the field of quantitative as well as in computer science. Probability is used in software engineering to assess the amount of risk. For example, suppose we are designing a system, and we are using probability. In this case, the probability will tell about the capacity of the system that means how much load our system can handle, and after that peak load, the system will crash. We can also measure the network's reliability using probability. In machine learning, we can do tasks ranging from developing good medical treatment to calibration of spam filters using various conditional probability applications. A randomized algorithm is known as the more efficient and best algorithm when it comes to practice because they provide the exact computing of those tasks that are difficult to compute. Probability can be described as one of the foundations of data science as well as statistics. It is also known as one of the hottest fields in the industry. If students are studying probability on the basis of computer science, it will provide them a quantitative intuition, and it is useful in their everyday life and throughout their careers.
When a developer develops any project, it is important that he should be confident of getting desired results by running their code. We can use mathematics to describe the programs. The reason for their correctness is propositional logic tools. The core area of computer science is known as algorithms, and it is difficult to analyze and design an algorithm by using these critical skills. The principle of mathematical induction is used by the two major paradigms: functional programming and iterative programming. This principle is used to verify their loops and recursive function calls separately. The most formal specification language can be called Logic used in the foundation and design of programming language. In the SQL family, languages are just the implementation of particular logical calculus, for instance. It is the same as the implementation of relational logic, which contains a lot of other domain-specific languages and many added features. In industry, there is an increment of adoption of formal methods and program verification. It is also used in tandem with techniques of traditional testing to increase confidence about the performance and effectiveness of the software.
Induction and Recursion
In the field of computer security, cryptography, and blockchain, the critical applications are contained by the Number theory. According to mathematics, the user's data is perfectly secured from the various types of attacks and malicious adversaries with the help of a modern cryptographic system. The mathematical basis for hashing is described by modular arithmetic, and it is the most useful tool for several applications. The files which are transferred by the internet are verified by the Checksum, and it is based on hashing. Data structures like hash map performs efficient operations by using modular arithmetic. In the operating system and computer architecture, number theory also provides the facility to use memory-related things.
We can also develop quantitative intuitions by using Counting techniques. For example, suppose the users create passwords by using some defined set of rules. Now we can get the number of valid passwords by using the counting technique. This technique is also used to determine the time duration taken by an attacker to brute force all the passwords. Now we will learn the pigeonhole principle, which describes why we don't have an algorithm that can describe universal lossless compression. When we use a compression algorithm, it decreases certain files every time and increases the other files. Different types of files, such as video, audio, text, images, etc, can be compressed by using each compression algorithm. The complexity of algorithm can be easily determined with the help of counting. The real-world application has a lot of different available resources that have a complicated tradeoff. Some tasks don't have a lot of space that's why they have to sacrifice their time for more space, while others require a fast algorithm because they can afford a huge space to achieve the speed. In a complex situation, we require to achieve a sweet spot in resource usages so that the system does not face the problem related to resource starvation and keeps running perfectly. Using counting, we are able to create these considerations in a structural manner. It can also provide a formal guarantee related to resource usage.
Graph is known as a powerful data structure. It has the ability to answer questions and model the relationship. For example, when we use our navigation app to search for the fastest route from our office to our home, this app uses the graph search algorithm to search it. It will also show us the time during according to our vehicle. The graph is extensively used in computer science to represent a file system. It is also used in database, deep learning, functional programming, and other applications.