# Discrete Mathematics

**Discrete Mathematics** deals with the study of Mathematical structures. It deals with objects that can have distinct separate values. It is also called Decision Mathematics or finite Mathematics. It is the study of mathematical structures that are fundamentally discrete in nature and it does not require the notion of continuity.

Objects that are studied in discrete mathematics are largely countable sets such as formal languages, integers, finite graphs, and so on. Due to its application in Computer Science, it has become popular in recent decades. It is used in programming languages, software development, cryptography, algorithms etc. Discrete Mathematics covers some important concepts such as set theory, graph theory, logic, permutation and combination as well. In this article, let us discuss these important concepts in detail.

## Discrete Mathematics Topics

**Set Theory: ** Set theory is defined as the study of sets which are a collection of objects arranged in a group. The set of numbers or objects can be denoted by the braces {} symbol. For example, the set of first 4 even numbers is {2,4,6,8}

**Graph Theory: **It is the study of the graph. The graph is a mathematical structure used to pair the relation between objects. Graphs are one of the prime objects of study in Discrete Mathematics.

**Logic:** Logic in Mathematics can be defined as the study of valid reasoning. There are three types of logic gates. They are AND(∧), NOT(~), and OR(∨)

**Permutation:** The different arrangements that can be made with a given number of sets taking some or all of them in a particular sequence at a time are called Permutation. For example, there are six permutations of the set {5,6,7}, namely (5,6,7), (5,7,6), (6,5,7), (6,7,5), (7,5,6), and (7,6,5).

**Combination:** The selection of a number of objects taking some or all of them at a time is called combination. The order of selection does not matter for the combination.

**Sequence:** According to some definite rules, a set of numbers arranged in a definite order is called a Sequence. A sequence is a function whose domain is the countable set of natural numbers.

**Series:** A series is the sum of the terms of a sequence. The result of adding all the terms together: s_{1}+s_{2}+s_{3}+s_{4}… is the sum of the series.

### Discrete Mathematics Applications

- The research of mathematical proof is especially important in logic and has applications to automated theorem demonstrating and regular verification of software.
- Partially ordered sets and sets with other relations have uses in different areas.
- Number theory has applications to cryptography and cryptanalysis.

### Discrete Mathematics Examples

**Example 1: Determine that in how many ways can three prizes be shared among 4 boys when**

**i) No one gets more than one prize.**

**ii) A boy can get any number of prizes.**

**Solution:**

i) The first prize can be given in 4 ways as one cannot get more than one prize, the remaining two prizes 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 prize can be given in 4 ways.

The total number of ways = 4^{3} = 64.

**Example 2:** **Find the sum of all four-digit numbers formed by using 2, 3, 6, 9 in which no digit is repeated.**

**Solution:**

If 2 occupies unit’s place, the remaining 3 digits can be arranged in 3!= 6 ways. Similarly, if 2 occupies ten’s place, hundreds place, thousand’s place, in each of these cases we get 3! numbers. Thus, the positional value contributed by 2 to the sum when it occupies different

values.

(3!)(2) + (3!)(20) + (3!)(200) + (3!)(2000) = 3!(2)(1111)

Similarly, the values contributed by 3, 6, 9 to the sum are

3! (3)(1111), 3! (6) (1111), 3! (9)(1111) respectively.

The required sum is 3!(1111)(2 + 3 + 6 + 9) = 1,33,320

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