Continuous Mathematics deals with structures that range in value, over real numbers like algebra, calculus, and statistics, etc.

 Discrete Mathematics and its application deals with the study of mathematical structures that is countable or otherwise distinct.

Let’s make it simple

Wristwatches can better illustrate the difference between continuous and discrete Mathematics.

Continuous mathematics resembles analog watches that separate hour, minute, and second hands.

For an analog watch, in between 10.02 am, and 10.03 am, there are infinite possible different times as the second-hand move around the clock.

Continuous Mathematics, similarly, studies the concepts of mathematics that are infinite in scope.

Discrete Mathematics resembles a digital watch that does not acknowledge split seconds.

There is no time between 10.02 am, and 10.03 am, the text leaps from one second to the next.

Digital watch shows one finite different time, and the transition from one time to the next is sharp and ambiguous.

Integers are the primary tools for discrete mathematics, while real numbers make Continuous Mathematics.

Discrete Mathematics is the language of computer science, and its relevancy is increasing every day.

Discrete Mathematics can be counted, placed into sets and put into ratios with one another.

Independent events along with the rules of Product, Sum and, PIE are common among Combinatorics, Set theory, Probability along with De Morgan’s Law in discrete mathematics.

Discrete Mathematics and its applications are challenging and exciting because of the restrictions placed on them.

It is rare in a discrete solution that a problem fits a specific formula.

Creatively applying different approaches to the problem make discrete mathematics all the more exciting.

The members of the discrete set include (but not limited to):

  • Relations and functions
  • Set theory
  • Logic and Boolean algebra
  • Sequences and series
  • Algorithms and theory of computation
  • Matrix theory
  • Number theory
  • Graph theory
  • Coding
  • Counting and discrete Probability and many other

Let’s look at the five main topics that make discrete Mathematics and its applications more relevant.

Combinatorics

Combinatorics is a branch of Mathematics that concern with the arranging and counting of things that are too large to be estimated the conventional way.

Combinatorics is concerned with how things are arranged and count possible outcomes in uniform Probability.

The “rule of product” and the “rule of sum” govern the counting arrangements of Combinatorics.

Combinatorics is especially useful in computer science to estimate how many operations a computer algorithm may require.

The subset of Combinatorics includes:

  1. Permutation: It is a collection of different types of arrangements of objects regarding an order.
  2. Combination: It is the arrangement of the object without order.
    This is different from permutation, where the order in arrangements matters.
  3. Derangements: A permutation in such a way that no object is in its original place of order.
  4. Rectangular Grid walks: Determining the number of paths through which a rectangular lattice can be crossed.
  5. Distribution of Objects into Bins: deciding how objects can be bundled into bins.

Combinatorics has various applications in the areas of Graph Theory, coding, cryptography, and Probability.


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Set Theory

Set theory is the branch in mathematics that deals with the formal collection of sets as units, without regard to the nature of the individual constituents.

Set theory serves as the foundation to which the rest of mathematics is built upon.

A “set” is a collection of specific contents enclosed in curly brackets {}

The order inside the set is not essential.

E.g. a set containing {1, 2, 3} is same as {3, 1, 2}

A set that has no elements is called an “empty” set, and it is denoted by ∅

At a basic level, set theory is about how the elements can be arranged, combined, and counted.

Some sub-sets to the Set theory include:

  1. Cardinality: It is the total elements given within a set.For a given Set A, Cardinality of that particular set is denoted by |A|
  2. Complement: A complement of a set A is the set of elements that are not in A but may belong to another Set.
  3. Union and Intersection: it presents possible outcomes of how the set can be combined.
  4. De Morgan’s law: The law postulates how opposites of a proposition or statement can relate mathematical statements and concepts.

De Morgan’s Law relates “intersection and union of Sets” or “Conjunction and disjunction of propositions” through “complements” or “Negation.”

Graph Theory

In Graph theory, a graph is a set of the structure of connected Nodes, which are, in some sense related.

The components that identify a graph are:

  • A node or a vertex (V)
  • And set of edges (E) that works as the connection between two nodes.
  • Each edge has either one or more associated vertices called “Endpoints.”

The different types of graphs in discrete mathematics are as follows:

  • Null graph: It is an empty graph where there are no edges between vertices.
  • Direct graph: The edges are directed by arrows to indicate the relationship between one vertex and another.
  • Undirected graph: The graph whose edges are not directed.
  • Connected graph: The graph where a path of edges connects every pair of vertices.
  • Disconnected graph: The graph where a path of edges does not connect every pair of vertices.
  • Bipartite graph: The graph that can be separated into two sets so that edges can go between sets but not within it.
  • Simple graph: The graph that has at most one edge between two pairs of vertices.
  • Multigraph: The graph has multiple edges between any given pair of vertices.
  • Planar graph: The graph that can be illustrated with at least one pair of its edge crossing.
  • Non-planer graph: The graph that cannot be illustrated with at least one pair of its edge crossing.

Discrete Probability

Discrete Probability is based on distinct sets of outcomes.

The probabilistic rule of product, sums, and complements work similar to the law of Combinatorics.

Many rules of Probability are relatable to the law of Combinatorics.

A discrete probability distribution takes a binary outcome as an assertion and gives Probability as a result.

Some discrete probability distributions that may be used in problems are:

  • Geometric distribution: Gives the Probability of the first success that will occur after a certain number of trials.
  • Binominal distribution: Gives the Probability of success after a certain number of trials.
  • Poisson distribution: Gives the Probability of an event that will occur after a specific number of times.

Symbolic Logic

Symbolic or formal Logic is the study of meaning and understanding of statements used to express mathematical ideas.

Some core concepts include:

  • Proposition: The sentences which can either be true or false.
  • Tautology: the proposition that is always true.
  • Contradiction: the proposition that is always false.
  • Compound proposition: Proposition built by the use of connectives “and,” “or” and “Not.”

Additionally, if two propositions “a” and “b” respectively arise, then the above compound connectives may give rise to:

  • a b, the conjunction meaning “a and b.”
  • a b, the disconjuction meaning “a or b.”
  • b, negation of b, meaning “not b.”

There is the possibility of not one but many Logic.

Logic is the pillar of computer science.

It is used every day to specify and reason about the properties of programs, algorithms, the state of applications, and solutions to programming problems.

What is calculus to natural science and engineering, Logic is to computer science.

Applications of discrete Mathematics:

While continuous mathematics presents in a massive operation, discrete Mathematics and its applications, however, remains ominous.

Yet, the applications of discrete mathematics form the basis of real-world scientific fields, such as:

Cryptography

Cryptography is the study of security structure and passwords for computers and other electronic transactions.

The entire process follows the principles of discrete mathematics as it allows cryptographers to break information into distinct bits.

As most financial and informational transaction happens online, there is a lot of security involved to protect just valuable assets.

Therefore, cryptographers must provide strong knowledge in number theory, an essential aspect of discrete mathematics, to provide a secure password and encryption programs.

Relational database

Relational databases work on the concept of “Sets.”

The organization relies on a relational database to keep track of its employees, client, and business records.

Through the aspect of Sets, the database groups’ relevant information of the organization, it can be a list of employees, clients, and business transactions, and link pertinent information to that group.

Sets help in keeping information grouped and in order.

Logistics

Logistics would have been obsolete without discrete mathematics.

Logistics make heavy use of graph theory, a subsidiary of discrete mathematics, to simplify the organization’s flow of information.

A logistician can use graph theory to simplify and analyze practical, logistical problems such as finding the best route for shipping a product.

Computer Algorithms

A computer operates on the rules of the algorithm.

Discrete mathematics and its application are most profound in computer programming.

Algorithms follow the rules of discrete mathematics to create an efficient design.

Because of discrete mathematics, we can speed up algorithmic steps for efficient outcomes.

Discrete mathematical applications have made computers run faster and perform better than ever.

If you are looking to know more about Discrete Mathematics, please visit our MyMathLab Answers page for useful insights.


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