CHAPTER 1 - Introduction to Advanced Discrete Structures

Discrete

means separate or divided.

discrete unit

is a separate part of something larger.

Discrete Mathematics

It is the branch of mathematics that deals with the object that can consider only distinct, separated values.

Discrete Mathematics

is also known as the language of computer science.

set

is a collection of distinct elements or objects. These elements can be anything-numbers, letters, people, etc.

Mathematical logic

is a branch of mathematics that deals with the study of formal systems for reasoning and the manipulation of symbols according to certain rules.

Mathematical logic

It includes topics like propositional logic, first-order logic, predicate calculus, and more. It is used as a foundation for rigorous mathematical reasoning and proof

Boolean algebra

is a branch of mathematics that deals with operations on sets of binary (two-valued) variables. It is used in digital circuit design, computer science, and other fields.

Boolean algebra

includes operations like AND, OR, and NOT, which correspond to logical operations of conjunction, disjunction, and negation

Graph theory

is the study of graphs, which are mathematical structures consisting of nodes (vertices) and edges that connect pairs of nodes. It has applications in various fields such as computer science, social networks, transportation networks, and more.

Graphs

are used to model relationshipsbetween objects or entities.

tree

is a hierarchical data structure with a root node and zero or more child nodes. It is used to represent hierarchical relationships, such as in file systems or data organization. Special types of it have applications in searching, sorting, and other algorithms

Group theory

is a branch of abstract algebra that deals with the study of &symmetry and structures* called groups. It has applications in physics, chemistry, cryptography, and other areas

group

consists of a set of elements along with an operation (usually multiplication or addition) that satisfies certain properties.

Probability

is a measure of the likelihood of an event occurring. It ranges from 0 (impossible event) to 1 (certain event).

Probability theory

is used to model uncertainty and randomness and is a fundamental concept in statistics, gambling, decision theory, and more.

Combinatorics

Counting theory, also known as _, deals with counting and arranging objects in various ways. It includes topics like permutations, combinations, and the binomial theorem. It is used in probability, statistics, and various fields where counting arrangements is important

Relation theory

involves the study of relationships between elements in sets.

relation

is a set of ordered pairs, and it can represent concepts like equality, inequality, or any other kind of connection between elements. It is used in various mathematical and computer science contexts, including databases and graph theory

Graph Theory

_ in computer Science is used to model pair-wise relations between objects from a certain collection

Combinatorics

is especially useful in computer science. _ methods can be used to develop estimates about how many operations a computer algorithm will require.

Combinatorics

_ is also important for the study of discrete probability. Such methods can be used to count possible outcomes in a uniform probability experiment.

Algorithm analysis

is an important part of computational complexity theory, which provides theoretical estimation for the required resources of an algorithmto solve a specific computational problem.

Discrete probability distribution

is a type of probability distribution that shows all possible values of a discrete random variable along with the associated probabilities. In other words, it gives the likelihood of occurrence of each possible value of a discrete random variable

probability distribution

is the mathematical function that gives the probabilities of occurrence of possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events

recurrence relation

is an equation which represents a sequence based on some rule. It helps in finding the subsequent term (next term) dependent upon the preceding term (previous term). If we know the previous term in a given series, then we can easily determine the next term.