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Introduction to Discrete Mathematics

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Experiments which are performed and measured according to the discrete values or continuous values both have different perceptions. Introduction to discrete mathematics is the branch of mathematics that refers to objects having only distinct and separate values or it can be said that this deals with the countable sets. Graph theory of hyper graphs, coding, the design of blocks, combinatorial and discrete geometry, matrices etc. fields are covered by discrete mathematics.
  
Basically, while giving discrete mathematics introduction, one cannot get the values on all the points as the function could be defined on some particular values only. But, continuous mathematics shows the combined result of smooth variation.

Congruence and recurrence relation are also a part of discrete mathematics.

To have a better understanding of these topics, the study of algorithms is must. The set of objects in discrete mathematics can have infinite of finite numbers. The removal of the errors is most significant in the discrete mathematics as compared with the continuous mathematics. In computer science, the areas of discrete mathematics are drawn on graph theory and logic. Besides that, in logic, combinatorics, probability, algebra, geometry and in all the sections, discrete mathematics concept is a must. Discrete is just the reverse process of continuous and the subject of engineering, digital signal processing and signal systems are totally based on this concept. Discrete mathematics is a contemporary field of mathematics that is widely used in business and industry. It is sometimes called the mathematics of computers.

Discrete Mathematics Definition

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Discrete mathematics is the mathematics that deals with discrete objects as the discrete objects are separated from each other. Here, the data can only take certain values. Data is discrete, if there are only a finite number of values possible and occurs in a case where there are only a certain number of values, or when we are counting something.

For example, to obtain taxi license in Las Vegas, a person must pass a written exam regarding different locations in the city. Number of attempts a person takes to attempt the test is an example of discrete data. A person could take it once, twice, and so on. The possible values are 1, 2, 3, ......

Topics in Discrete Mathematics

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The topics of discrete mathematics are given below:

Algorithm: A specific set of instructions for carrying out a procedure or solving a problem, usually with the requirement that the procedure terminates at any point.

Counting: Using mathematical techniques to systematically count objects.

Deviation: Difference between the value of an observation and the mean of the population.

Matrices: Rectangular arrays of numbers. Used in algebra, geometry, statistics, and probability, with many applications in science, business, and industry.

Permutation: A rearrangement of the elements in an ordered list S into a one-to-one correspondence with S itself.

Combinatorics: Branch of mathematics concerning the study of finite or countable discrete structures.

Graph: A graph that contains no cycles.

Graph Theory: The study of formal mathematical structures called graphs.

Ratio: A ratio is a relationship between two numbers of the same kind. Usually expressed as A : B.

Proportion: A name given to a statement, when the two ratios are equal. It can be written using a colon, a : b = c : d

Polar coordinates: This is the system of coordinates in which coordinates are defined with the help of angles.

Probability: Branch of mathematics that deals with the likelihood that a particular event will happen in the future. It is expressed as a number between 1 and 0 and can be expressed in fractions, ratios, percentage.

Discrete Mathematics Problems

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Given below are some of the problems in discrete mathematics.

Solved Examples

Question 1: Find the fifth perfect prime number.
Solution:
First five prime numbers are 2, 3, 5, 7 and 11
The fifth prime number is 11
The formula to find perfect prime number is (2$^{p-1}$) (2$^{p}$ - 1)

Plugging in 11 in the above formula, we get
(2$^{11-1}$) (2$^{11}$ - 1)
= 4096 x 8191
= 33550336

Question 2: Find the value of x in the 4 : 3 = x : 9
Solution:
We can write 4 : 3 = x : 9 as $\frac{4}{3}$ = $\frac{x}{9}$
4 × 9 = x × 3
x × 3 = 4 × 9
x = $\frac{4× 9}{3}$
x = 12

Question 3: When a die is rolled, what is the probability of obtaining an even number?
Solution:
When a die is rolled, the possible outcomes are 1, 2, 3, 4, 5 and 6.
Total number of sides = 6
Number of ways to roll an even number = {2, 4, 6} = 3

P(Even) = $\frac{\text{Number of ways to roll an even number}}{\text{Total number of sides}}$ = $\frac{3}{6}$

Probability of obtaining an even number is $\frac{1}{2}$