Chapter 1_Introduction to Advanced Discrete Structures (copy)

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.

language of computer science

Discrete Mathematics is also known as the _____

latter half of the twentieth century,

digital computers

Research in discrete mathematics increased in the _____ partly due to the development of _____ which operate in "discrete" steps and store data in "discrete" bits.

computer algorithms ,

programming languages,

cryptography,

automated theorem proving,

software development

Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science, such as (5)

computer implementations

are significant in applying ideas from discrete mathematics to real-world problems.

Set,

Mathematical Logic,

Boolean Algebra,

Graph Theory,

Trees,

Group Theory,

Probability,

Recurrence Relations,

Counting Theory,

Relation Theory

Foundations of Discrete Mathematics (9)

set

is a collection of distinct elements or objects.

set

These elements can be anything - numbers, letters, people, etc.

set

are a fundamental concept in mathematics and are used to define many other mathematical concepts.

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.

Mathematical logic

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.

Boolean algebra

It is used in digital circuit design, computer science, and other fields.

AND, OR, NOT

Boolean algebra includes operations like _____, _____, and _____ 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.

Graph theory

are used to model relationships between objects or entities.

Graph theory

has applications in various fields such as computer science, social networks, transportation networks, and more.

tree

is a hierarchical data structure with a root node and zero or more child nodes

tree

used to represent hierarchical relationships, such as in file systems or data organization.

binary trees, balanced trees

Special types of trees, like _____ and _____, 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.

group

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

Group theory

_____ has applications in physics, chemistry, cryptography, and other areas.

Probability

is a measure of the likelihood of an event occurring.

Probability

It ranges from 0 to 1.

impossible event

In probability, 0 means _____

certain event

In probability, 1 means _____

Probability theory

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

combinatorics

Counting theory is also known as _____

Counting theory

deals with counting and arranging objects in various ways. It includes topics like permutations, combinations, and the binomial theorem.

Counting theory

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.

relation

used in various mathematical and computer science contexts, including databases and graph theory.

graph theory,

counting and combinatorics,

analysis of algorithms,

discrete probability,

recurrence relations

Uses of Discrete Structures in Computer Science (5)

Graph Theory

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

computer science

• Combinatorics is especially useful in _____.

• Combinatorics methods can be used to develop estimates about how many operations a computer algorithm will require.

discrete probability

• Combinatorics is also important for the study of _____.

• Combinatorics 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 algorithm to solve a specific computational problem.

computational complexity theory

provides theoretical estimation for the required resources of an algorithm to 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.

Discrete probability distribution

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.

probability distribution

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.

recurrence relation

It helps in finding the subsequent term (next term) dependent upon the preceding term (previous term).

recurrence relation

recurrence relation

standard pattern

• Since a _____ is developed now, we can find the set of new terms.

  • This is also applicable for arithmetic and geometric sequence.