Discrete Mathematics

Mathematics is a subject that you’ll either love or dread. It is as simple as that. The people who dread Mathematics are the ones who have not witnessed the beauty of numbers and logic. So, remember it’s never too late for absorbing knowledge. This subject not only teaches us how to deal with problems but also instills common sense in us. Mathematics is one of the subjects which can never truly and entirely separate from our lives. The concepts of Mathematics serve as the basis of various other subjects like physics, computer science, architecture etc. Mathematics is divided into 4 branches namely, arithmetic, algebra, geometry, and trigonometry. Did you know that Archimedes is considered as the Father of Mathematics? Today we’ll learn about Discrete Mathematics.

Do you know what Discrete Mathematics is? Do you know about Discrete Mathematics and its applications? We’ll discuss it all here.

Discrete Mathematics

Discrete Mathematics is about Mathematical structures. It is about things that can have distinct discrete values. Discrete Mathematical structures are also known as Decision Mathematics or Finite Mathematics. This is very popularly used in computer science for developing programming languages, software development, cryptography, algorithms, etc.

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Discrete Mathematics Problems and Solutions

Now let’s quickly discuss and solve a Discrete Mathematics problem and solution:

Example 1: 

Determine in how many ways can three gifts be shared among 4 boys in the following conditions-

i) No one gets more than one gift.

ii) A boy can get any number of gifts.

Solution:

i) The first gift can be given in 4 ways as one cannot get more than one gift, the remaining two gifts can be given in 3 and 2 ways respectively.

The total number of ways = 4 x 3 x 2 = 24.

ii) As there is no restriction, each gift can be given in 4 ways.

The total number of ways = 43 = 64.

Discrete Mathematics Topics

Various concepts of Mathematics are covered by Discrete Mathematics like:

1. Set Theory

2. Permutation and Combination

3. Graph Theory

4. Logic

5. Sequence and Series

1.  Set Theory

Set Theory is a branch of Mathematics that deals with collection of objects. It starts with the fundamental binary relation between an object M and set A.

Imagine there are two sets, say, set A and set B. Set A has numbers 1-5 and Set B has numbers 1-10. You can see that all the elements of set A are in set B. This makes set A a subset of set B.

A={1,2,3,4,5}          B={1,2,3,4,5,6,7,8,9,10}

Imagine there are two sets, say, set A and set B. Set A has numbers 1-5 and Set B has numbers 1-10. If we combine the elements of set A and set B, then the set we get is called a union set. So, we get the union of set A and set B. 

(AUB)={1,2,3,4,5,6,7,8,9,10}

Imagine there are two sets, say, set A and set B. Set A has numbers 1-5 and Set B has numbers 1-10. If we take the elements that are present in both sets then we get the intersection.

(A∩B)={1,2,3,4,5}

Imagine there are two sets, say, set A and set B. Set A has numbers 1-5 and Set B has numbers 1-10. When we are trying to find the Cartesian Product of set A and B, we are actually making an ordered pair. (AXB)={(1,1);(1,2).........(5,4);(5,5)}

Remember  (AXB)≠(BXA)

2. Permutation and Combination 

Permutation and Combination are all about counting and arranging from the given data. The permutation is all about arranging the given elements in a sequence or order. The combination is about selecting elements in any way required and is not related to arrangement.

You can use the formula for permutation – nPr = \[\frac{(n!)}{(n-r)!}\]

Where r objects have to be arranged out of a total of n number of objects

The formula for combination is—  \[nCr=\frac{n!}{r!(n-r)!}\]

 Where r objects have to be chosen out of a total of n number of objects 

3. Logic 

Logic can be defined as the study of valid reasoning. The logical formulas are discrete structures and so are proofs thus, forming finite trees. The truth values of logical formulas form a finite set. They are restricted to only two values either true or false. 

4. Graph Theory

Graph Theory is about the study of graphs. They are discrete Mathematical structures and are used to model in relation to pairs between the objects. The graph we are discussing here consists of vertices which are joined by edges or lines. Graphs are one of the most important objects of study in Discrete Mathematics. Discrete Mathematics and graph theory are complementary to each other. Graphs are present everywhere. They are models of structures either made by man or nature. They can model various types of relations and process dynamics in physical, biological and social systems. They can also display networks of communication, data organization, the flow of computation, etc. they are also used in geometry and in topology. 

5. Sequence and Series

A sequence is a set of numbers which are arranged in a definite order and following some definite rule. A series is a sum of terms which are in a sequence.

Formulas Related to Some Special Series

1.  The sum of 1st n natural numbers: 

    \[S_{n} = \frac{n(n+1)}{2}\] 

2.  The sum of the squares of 1st n natural numbers:  

       \[S_{n} = \frac{n(n+1)(2n+1)}{6}\]

3.  The sum of the cubes of first n natural numbers:  

         \[S_{n} = \text{(Sum of the first n natural numbers)}2\]

    \[= \frac{n(n+1)}{22}\]

Fun Facts

• On contrary to real numbers that differs "seamlessly", Discrete Mathematics studies objects such as graphs, integers and statements in reasoning

• The objects studied in Discrete Mathematics do not differ seamlessly, in fact, have varied, Discrete Mathematics does not include matters in "continuous mathematics" such as algebra and calculus          

Continuous and Discrete Mathematics

Mathematics can be divided into two categories: continuous and discrete.

Continuous Mathematics is based on a continuous number line or real numbers in continuous form. It is defined by the fact that there is virtually always an endless quantity of numbers between any two integers. In continuous Mathematics, for example, a function can be depicted as a smooth curve with no breaks.

Discrete Mathematics involves separate values; that is, there are a countable number of points between any two points in Discrete Mathematics. If we have a finite number of items, for example, the function can be defined as a list of ordered pairs containing those objects and displayed as a complete list of those pairs.

Use of Discrete Mathematics in Real World 

It is commonly stated that Mathematics may be used to solve a wide range of practical problems. Cathy and MathILy-Er focus on Discrete Mathematics, which supports nearly half of pure Mathematics, operations research, and computer science in general. Discrete Mathematics is becoming more prevalent in academia and industry as time goes on. But what exactly are the applications that people are referring to when they claim Discrete Mathematics can be used? What issues are being addressed? This article attempts to answer those questions. There are summaries of Discrete Mathematics applications in our daily lives, as well as in major and interesting research and corporate applications, with links to extended explanations.

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