Discrete Mathematics PDF Summary
Discrete mathematics is the study of mathematical structures that are considered not “continuous” (analogous to continuous functions) but “discrete” (analogous to discrete variables, consisting of a binomial with a set of natural numbers). Objects analyzed in discrete mathematics include integers, graphs, and words in logic. In contrast, discrete mathematics excludes topics in “continuous mathematics” such as real numbers, calculus, or Euclidean geometry. Discrete objects can often be quantified by integers; More formally, discrete mathematics is classified as the branch of mathematics dealing with countable sets (finite sets or sets with the same cardinality as the natural numbers). Nevertheless, the term “discrete mathematics” accomplishes not have a precise definition
The set of objects studied in discrete mathematics may be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deal with finite sets, especially those related to business.
Discrete Mathematics PDF- Topics in discrete mathematics
Theoretical computer science
Complexity studies the time taken by algorithms, such as this sorting routine.
Computational geometry applies computer algorithms to representations of geometrical objects.
Theoretical computer science includes areas of discrete mathematics relevant to computing. It draws heavily on graph theory and mathematical logic. Included within theoretical computer science is the study of algorithms and data structures. Computability studies what can be computed in principle, and has close ties to logic, while complexity studies the time, space, and other resources taken by computations.
Information theory
The ASCII codes for the word “Wikipedia”, given here in binary, provide a way of representing the word in information theory, as well as for information-processing algorithms.
Information theory involves the quantification of information. Closely related is the coding theory which is used to design efficient and reliable data transmission and storage methods. Information theory also includes continuous topics such as analog signals, analog coding, and analog encryption.
Logic
Logic is the study of the principles of valid reasoning and inference, as well as consistency, soundness, and completeness. For example, in most systems of logic (but not in intuitionistic logic) Peirce’s law (((P→Q)→P)→P) is a theorem. For classical logic, it can be easily verified with a truth table. The study of mathematical proof is particularly important in logic and has applications to automated theorem proving and formal verification of software.
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects, such as {blue, white, red} or the (infinite) set of all prime numbers. Partially ordered sets and sets with other relations have applications in several areas.
In discrete mathematics, countable sets (including finite sets) are the main focus. The beginning of set theory as a branch of mathematics is usually marked by Georg Cantor’s work distinguishing between different kinds of infinite sets, motivated by the study of trigonometric series, and further development of the theory of infinite sets is outside the scope of discrete mathematics.
Combinatorics
Combinatorics studies the way in which discrete structures can be combined or arranged. Enumerative combinatorics concentrates on counting the number of certain combinatorial objects – e.g. the twelvefold way provides a unified framework for counting permutations, combinations, and partitions. Analytic combinatorics concerns the enumeration (i.e., determining the number) of combinatorial structures using tools from complex analysis and probability theory.
Graph theory
Graph theory has close links to group theory. This truncated tetrahedron graph is related to the alternating group A4.
Graph theory, the study of graphs and networks, is often considered part of combinatorics but has grown large enough and distinct enough, with its own kind of problems, to be regarded as a subject in its own right.[19] Graphs are one of the prime objects of study in discrete mathematics. They are among the most ubiquitous models of both natural and human-made structures.
Number theory
Number theory is concerned with the properties of numbers in general, particularly integers. It has applications in cryptography and cryptanalysis, particularly with regard to modular arithmetic, diophantine equations, linear and quadratic congruences, prime numbers, and primality testing. Other discrete aspects of number theory include the geometry of numbers.
Algebraic structures
Algebraic structures occur as both discrete examples and continuous examples. Discrete algebras include: boolean algebra used in logic gates and programming; relational algebra used in databases; discrete and finite versions of groups, rings, and fields are important in algebraic coding theory; discrete semigroups and monoids appear in the theory of formal languages.
Discrete analogues of continuous mathematics
There are many concepts and theories in continuous mathematics which have discrete versions, such as discrete calculus, discrete Fourier transforms, discrete geometry, discrete logarithms, discrete differential geometry, discrete exterior calculus, discrete Morse theory, discrete optimization, discrete probability theory, discrete probability distribution, difference equations, discrete dynamical systems, and discrete vector measures.
Calculus of finite differences, discrete analysis, and discrete calculus[edit]
In discrete calculus and the calculus of finite differences, a function defined on an interval of the integers is usually called a sequence. A sequence could be a finite sequence from a data source or an infinite sequence from a discrete dynamical system. Such a discrete function could be defined explicitly by a list (if its domain is finite), by a formula for its general term, or it could be given implicitly by a recurrence relation or difference equation.
Discrete geometry
Discrete geometry and combinatorial geometry are about combinatorial properties of discrete collections of geometrical objects. A long-standing topic in discrete geometry is a tiling of the plane.
In algebraic geometry, the concept of a curve can be extended to discrete geometries by taking the spectra of polynomial rings over finite fields to be models of the affine spaces over that field and letting subvarieties or spectra of other rings provide the curves that lie in that space. Although the space in which the curves appear has a finite number of points, the curves are not so many sets of points as analogues of curves in continuous settings.
Discrete modelling
In applied mathematics, discrete modelling is the discrete analog of continuous modelling. In discrete modelling, discrete formulae are fit to data. A common method in this form of modeling is to use recurrence relation. Discretization concerns the process of transferring continuous models and equations into discrete counterparts, often for the purposes of making calculations easier by using approximations. The numerical analysis provides an important example.
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