Discrete Math

Summary

In this video, Suraj introduces the importance of discrete mathematics in computer science, showcasing its relevance through a cybersecurity scenario involving cracking a password-protected database. He explains the foundational concepts of discrete math, such as graph theory, algorithms, number theory, recursion, and set theory, and demonstrates their practical applications. Suraj also highlights the role of logic in creating proofs for algorithms, providing a comprehensive overview of how discrete math underpins computer science.

Takeaways

• 😀 Discrete mathematics is essential for coding and computer science, including areas like machine learning.
• 🔍 Discrete math is a collection of mathematical branches that deal with discrete rather than continuous objects.
• 📚 It is a fundamental part of computer science education and is included in most degree programs.
• 🛣️ Discrete math helps in designing networks, message routing paths, web searches, algorithm analysis, and cryptographic protocols.
• 🌐 Graph theory, a part of discrete math, is used to model relations between objects and find optimal paths, like the shortest path to a lab.
• 🔑 Dijkstra's algorithm is highlighted as a famous method in computer science for finding the shortest path in a graph.
• 🔒 The script discusses a scenario where a password-protected database needs to be accessed, requiring knowledge of discrete math to decrypt it.
• 🔢 The concept of brute-forcing a password is introduced, along with combinatorics to calculate the number of possible combinations.
• 🔄 Recursion is presented as a method in discrete math to solve problems by relying on solutions to smaller instances of the same problem.
• 🔡 The Caesar cipher, a type of encryption, is explained as an example of how characters can be shifted to encode and decode messages.
• 📊 Set theory, another aspect of discrete math, underlies relational databases and is used to compile reports from databases like SQL.
• 📝 Logic in discrete math is crucial for creating proofs and reasoning in computer science, with propositions as the basic building blocks.

Q & A

Why is discrete mathematics important for computer science?

Discrete mathematics is important for computer science because it provides the mathematical language needed to design high-speed networks, message routing paths, perform web searches, analyze algorithms for efficiency, and design cryptographic protocols, among other applications.

What is the difference between discrete and continuous mathematics?

Discrete mathematics deals with objects in discrete bundles, such as individual items or integers, while continuous mathematics deals with objects that vary without interruption, like measurements that can take on any value within a range.

How is graph theory related to the problem of finding the shortest path in a network?

Graph theory is used to represent a network as a graph with vertices and edges. It helps in designing algorithms, like Dijkstra's shortest path algorithm, to compute the fastest route by considering the weights of the edges, which represent the time or distance between nodes.

What is Dijkstra's shortest path algorithm, and how does it work?

Dijkstra's algorithm is a procedure used to find the shortest path between nodes in a graph. It starts by marking the initial node with a distance of zero and the rest with infinity, then iteratively selects the unvisited node with the smallest distance, updates the distances of its neighbors, and marks it as visited until all nodes are visited.

How does the concept of recursion help in solving the password decryption problem?

Recursion is a method where a function calls itself with a modified argument to solve a problem by breaking it down into smaller instances. In the case of the password decryption, a recursive function can be used to apply a shift operation to each character in the string until the entire phrase is decrypted.

What is the significance of combinatorics in the context of the encrypted password?

Combinatorics helps in calculating the total number of possible combinations for the encrypted password. It shows that with 36 characters to choose from for each of the 6 positions, there are over 2 billion possible combinations, indicating the need for a more efficient approach than brute force.

How does number theory contribute to understanding the relationships between numbers in cryptography?

Number theory involves analyzing the mathematical relationships between numbers, such as odd and even numbers, prime numbers, and sequences like the Fibonacci sequence. This understanding can help in deciphering encryption methods, like the Caesar cipher, which is based on shifting letters by a fixed number of positions in the alphabet.

What is a relational database, and how is it connected to set theory?

A relational database is a type of database that organizes data into tables with rows and columns, allowing for complex queries and operations. It is based on set theory, which deals with the study of collections of elements, and operations like intersections and unions that are fundamental to SQL queries.

What is the role of logic in mathematical reasoning and computer science?

Logic provides the rules for understanding and reasoning with mathematical statements. It is the basis for all mathematical reasoning and has practical applications in computer science, such as in the creation of proofs and the development of algorithms and programming languages.

How can the concepts of discrete mathematics be applied to real-world problems like cracking a password-protected database?

The concepts of discrete mathematics, such as graph theory, combinatorics, recursion, number theory, and logic, can be applied to real-world problems like password decryption by providing the mathematical framework for designing algorithms, analyzing possible combinations, and logically reasoning through the decryption process.

Outlines

00:00

📚 Introduction to Discrete Mathematics in Computer Science

This paragraph introduces the importance of discrete mathematics in computer science and coding. It explains that discrete math is not a traditional branch of mathematics but a collection of topics that deal with discrete rather than continuous quantities. The video aims to provide an overview of foundational topics in discrete math, using a scenario where the viewer plays a cybersecurity agent tasked with cracking a password-protected database. The paragraph sets the stage for the practical applications of discrete math in computer science, such as designing networks, performing searches, analyzing algorithms, and cryptographic protocols.

05:01

🛤️ Applying Graph Theory and Dijkstra's Algorithm

The second paragraph delves into the application of graph theory and Dijkstra's shortest path algorithm in finding the quickest route to a lab. It describes how streets and intersections can be represented as a graph to calculate the fastest path, emphasizing the undirected nature of the graph. The paragraph outlines the steps of Dijkstra's algorithm, from marking the initial node to updating distances and visiting nodes, and concludes with the successful use of a Python script to find the shortest path, demonstrating the practical use of discrete math in problem-solving.

10:04

🔐 Decrypting Passwords with Discrete Math Techniques

This paragraph discusses the process of decrypting an encrypted password using various discrete math concepts. It starts with the idea of brute-forcing all possible character combinations and then moves on to exploring number theory and the relationships between numbers. The paragraph introduces the concept of recursion, explaining it as a method to solve problems by relying on solutions to smaller instances of the same problem. It then describes how a Caesar cipher was used to encrypt the password and how recursion was employed to crack it by shifting characters in the alphabet, ultimately revealing the password.

📊 Set Theory and Logic in Database Analysis

The final paragraph covers the use of set theory and logic in analyzing a database to compile a report of the top 10 wealthiest criminals. It explains how relational databases, such as SQL, are based on set theory and how operations on sets, like intersections and unions, are fundamental to database queries. The paragraph also touches on the role of logic in mathematical reasoning, discussing propositions and how they form the basis for proofs. It concludes with the creation of a proof for the decryption algorithm, showcasing the integral role of discrete math in computer science and logical reasoning.

Keywords

💡Discrete Mathematics

Discrete Mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In the context of the video, it is presented as a crucial subject for computer science, covering topics that are not traditionally found in subjects like calculus or linear algebra. The video emphasizes its importance in fields such as designing networks, analyzing algorithms, and cryptographic protocols, illustrating its practical applications with the example of a cybersecurity agent cracking a password-protected database.

💡Graph Theory

Graph Theory is a branch of mathematics concerned with graphs, which are mathematical structures used to model pairwise relations between objects. The video uses the analogy of finding the shortest path to a lab to explain how graph theory can be applied to compute the fastest route on a network of streets represented as a graph. Dijkstra's shortest path algorithm is highlighted as a method to solve such problems within the realm of discrete mathematics.

💡Brute Force

Brute Force, in the context of the video, refers to the method of trying all possible combinations to find a solution, such as decrypting a password. The script mentions a brute-force script to try every combination of characters for a 6-character password, highlighting the impracticality of this approach given the vast number of combinations (over 2 billion), thus prompting the need for a more intelligent strategy.

💡Combinatorics

Combinatorics is the study of counting, both as a means and an end in obtaining results. The video uses combinatorics to calculate the total number of possible password combinations, demonstrating how it is part of discrete mathematics. It shows that with 36 characters to choose from for each of the 6 positions, the number of combinations is 36 to the power of 6.

💡Number Theory

Number Theory is the study of the properties and relationships of numbers, including odd and even numbers, prime numbers, and various sequences. In the video, the protagonist uses number theory to hypothesize that each character in the encrypted password could correspond to a number, leading to the discovery of a Caesar cipher used for encryption.

💡Recursion

Recursion in the video is introduced as a method of problem-solving where the solution depends on solutions to smaller instances of the same problem. It is exemplified by a recursive function used to decrypt the password by shifting characters in the string. The video humorously notes the self-referential nature of the term 'recursion' on Google.

💡Caesar Cipher

The Caesar Cipher is a simple encryption technique where each letter in the plaintext is replaced by a letter some fixed number of positions down the alphabet. The video uses this method as the encryption technique for the password, which is decrypted by applying a shift operation to each character in the encrypted string.

💡Relational Database

A Relational Database is a type of database that stores data in tables, which is based on set theory, a part of discrete mathematics. The video mentions using SQL queries to compile a report of the top 10 wealthiest criminals from a database, illustrating the application of set theory in the form of operations on sets within a relational database system.

💡Set Theory

Set Theory is the branch of mathematics that deals with sets, which are collections of distinct elements. The video explains how set theory underpins relational databases and SQL operations, such as inner joins, which are analogous to the intersection of sets in set theory.

💡Logic

Logic, as discussed in the video, is the study of the principles of correct reasoning and the criteria for the validity of arguments. It is used to create proofs, such as the proof of the decryption algorithm in the video, where the decrypted text is represented as a function of character shifts within a set of possible characters.

Highlights

Discrete mathematics is essential for coding and computer science, becoming increasingly important in recent years.

Discrete math is a collection of math branches that deal with discrete rather than continuous objects.

Graph theory, a part of discrete math, is used for designing high-speed networks and message routing paths.

Dijkstra's shortest path algorithm is a famous computer science algorithm for finding the fastest route in an undirected graph.

Brute-forcing a 6-character password with alphanumeric characters results in over 2 billion possible combinations.

Combinatorics, a branch of discrete math, helps in counting the number of possible outcomes in a scenario.

Number theory, which involves analyzing mathematical relationships between numbers, is key to decrypting passwords.

Recursion, a method in discrete math, involves functions calling themselves to solve problems based on smaller instances.

The Caesar cipher is a simple encryption method where each letter is replaced by a letter a fixed number of positions down the alphabet.

Set theory, foundational to relational databases like SQL, is based on the concept of sets and their operations.

SQL queries can compile reports by selecting top records based on specific criteria, utilizing set theory principles.

Logic in discrete math specifies the meaning of mathematical statements and helps in reasoning and proving algorithms.

Propositions and truth tables are basic building blocks of logic used to construct proofs in computer science.

The video provides a free, open-source textbook on discrete math for further study.

Discrete math topics such as recursion, logic, and combinatorics form the basis of computer science.

The video concludes with the importance of discrete math in various computer science applications.

Transcripts

discrete math will help us build trauma

hello world it's Suraj and if you're

interested in coding not just machine

learning but any kind of coding you need

to understand discrete mathematics it's

become a really important subject in the

past few years because of how useful it

is in computer science in fact pretty

much every single computer science

degree program contains at least one

course on discrete mathematics sometimes

to liberal arts majors it's okay breathe

I'm here now in this video I'm gonna

give you an overview of the foundational

topics in discrete math to help you

understand how it all works

learn about each of them by necessity

since we'll play the role of a

cybersecurity agent working for mi6 our

task is to crack a password-protected

database of international criminals

that's stored on a USB Drive discrete

math isn't actually the name of a

traditional branch of mathematics like

calculus or linear algebra is its

instead a description of a set of

branches of math that all have in common

one feature they're discrete rather than

continuous discrete stands for

individually separate and distinct while

continuous stands for forming an

unbroken whole without interruption

discrete math deals in objects in

discrete bundles like one or two

kendrick lamar albums at least three if

you're a real fan especially down in

contrast continuous math deals with

objects that vary continuously like

three point four centimeters from a wall

a good analogy would be digital watches

versus analog watches the ones where the

second hand loops around continuously

without stopping

basically discrete data is able to take

on only integer values but continuous

data can take on any value discrete math

was created several decades ago as the

mathematical language of computer

science colleges learned that

traditional math subjects did not cover

the type of math needed by computer

scientists so they put a bunch of math

topics together and called it discrete

math discrete math helps us design high

speed networks and message routing paths

perform web searches analyzed algorithms

for efficiency and design cryptographic

protocols that can do things like

automatically block any image of Wilson

as a genie beyond cringy before we can

start cracking the password to get the

data on the USB drive we've got to get

to the lab and there are a lot of

different paths we can take they drive

there but we're on a deadline so we want

to get to the lab as fast as possible

meaning we want to take the shortest

path to compute this we'll need to

represent the collection of streets and

intersections as a graph so that we can

design an algorithm that will calculate

the fastest way to get there since each

street is two-way this is an undirected

graph so there's no distinction between

the two vertices associated with each

edge this is graph theory the study of

graphs and an important part of discrete

math graphs are mathematical structures

used to model relations between objects

in our case intersections in graph

theory the problem of finding a path

between two vertices such that the sum

of the weights of its constituent edges

is minimized would be ideal the weights

in our case would be the time it takes

to drive from one vertex to the other

here we can use one of the most famous

algorithms in computer science

Dijkstra's shortest path algorithm which

is a five step procedure luckily we're

not in LA even Dijkstra is no match for

its traffic we start by marking our

initial node which is our current

location with the current distance of

zero and the rest with infinity then we

set the non visited node with the

smallest current distance as the current

node for each neighbor of our current

node we add the current distance of the

current node with the weight of the edge

connecting it to the neighbor if it's

smaller than the current distance of n

we set it as a new current distance of n

we then mark the current node as visited

and if there are any non visited nodes

left we repeat the process starting at

the second step until we visited all the

nodes thanks to a little Python script

we can easily find the shortest path to

the lab saving us a lot of time now that

we've arrived at the lab let's plug this

USB into the computer to see what we've

got here in order to access the database

we need a password and we have the

password here but it's encrypted we need

to figure out how to decrypt it into the

actual password the only thing we know

is that the length of the password is 6

characters long and it can contain

either numbers alphab

characters or both no emojis thankfully

our first strategy here will be to see

if we can write a script that will

brute-force all possible combinations of

every character a through Z and 1

through 9 but when we start running the

script it starts taking quite a while in

the meantime let's try and calculate how

many combinations are possible here

since there are 26 letters and 10 digits

we have a total of 36 characters to

choose from for each position if we

start from the left we can fill in the

first position 36 possible ways for each

of the 36 possibilities for the first

character there are 36 possibilities for

the second character that means our 36

squared waste fill in the first two

characters for each of the 36 squared

ways to fill in the first two characters

there are an additional 36 ways to fill

in the third character so there are 36

cubed ways to fill in the first three

characters notice the pattern here and

know that number 69 is not involved here

the total number of possible

combinations is 36 to the n where n is

the length of the character field which

is 6 in our case making it over 2

billion possible combinations way too

much to brute-force so we can go ahead

and pause our script this is an example

of common atour x the study of counting

both has a means and an end in obtaining

results and it's a part of discrete math

we need to think of a smarter way to

decode this password what if each of the

characters in the string what's

equivalent to a number and the actual

password was solely numerical what could

the mapping from characters and numbers

be here

maybe if we converted each character to

its numerical position in the alphabet

it would represent a prime number one

that's X counts away from zero but if we

try that out it doesn't work never may

be the numerical version of it correlate

to some sort of sequence here notice how

I'm trying to categorize these integers

into different numbered types based on

their relationships the study of

relationships between numbers is called

number theory a part of discrete math

there are odd numbers and even numbers

square numbers Pythagorean triples the

Fibonacci sequence number theory

involves analyzing the mathematical

relationships between numbers

right now I'm trying to figure out the

theory behind these numbers in

particular maybe each letter of this

text can actually be replaced by a

letter some fixed number of positions

down the alphabet the question then

becomes how many positions let's write

out a function for this one where we can

call the function inside of itself for

each character in the string we'll apply

a shift operation and recursively call

our function until we're done shifting

the entire phrase recursion a part of

discrete math is a way to solve a

problem where the solution depends on

solutions to smaller instances of the

same problem if we google search the

word recursion it recursively asks did

you mean recursion which is a fun little

easter egg from the programmers at

Google in fact the internet is riddled

with these a popular reddit threads top

answer for the question what is

recursion is a link to that very same

post an example of recursion a recursive

function is a function that calls itself

until some base condition is true and

the execution stops when we write out

our recursive function we can try out

different lengths of the possible shifts

until we get one that works and it turns

out that it's three shifts that was the

password we just needed to shift the

characters it turns out that this

password was encrypted using what's

called the Caesar cipher one of the

simplest methods of encryption each

letter of a given text is replaced by a

letter some fixed number of positions

down the alphabet for example with the

shift of one a would be replaced by B B

would become C and so on it's named

after Julius Caesar who apparently use

it to communicate with his officials is

this ambition this is Bridgeland

we now have the database of known

criminals this is a sequel database and

we'll want to compile a report for mi6

now that we've cracked it and that lists

the top 10 most wealthy criminals by

amount of money they donated to

underground organizations in order to do

that we'll write some sequel queries

which will select the top 10 highest

spenders from a certain section of the

database relational databases like

sequel are based almost entirely upon

set theory a part of discrete math as

is nothing more than an unordered

collection of elements with absolutely

no duplicates even the most complicated

sequel statements are nothing more than

operations on sets a sequel inner join

is the intersection of two sets for

example a Venn diagram is an easy way to

explain the concept of sets two

different sets have distinct values and

where they intersect they have similar

values unions disjunctions there are all

sorts of terminologies related to sets

and this is what set theory is all about

we can write down the top ten criminals

for our report but we have one more step

here we need to create a proof for our

report to show mi6 how we crack the code

the rules of logic a part of discrete

math specify the meaning of mathematical

statements they help us understand and

reason with statements like there exists

an integer that is not the sum of two

squares it's the basis of all

mathematical reasoning and has practical

applications in all of computer science

one of the basic building blocks of

logic are propositions a declarative

sentence that is either true or false

but not both like DC is the capital of

the USA is true while DC is the capital

of Mexico is false negations truth

tables combinations conjunctions we can

build off of the basic rules of logic to

create all sorts of proofs so logically

speaking we can define the proof of our

algorithm with the following equation

where the decrypted version of the text

is equal to each character shifting n

degrees out of 26 possible characters

mi6 is going to be very pleased with

this we'll probably get a double ho

status now license code so those are

some of the major concepts in discrete

math I've listed them all in the video

description as well as a free

open-source textbook I found on the

topic of discrete math study it there

are three things to remember from this

video discrete math is a set of branches

of math that all have in common the

feature that they are discrete rather

than continuous discrete data can only

take on integer values whereas

continuous data can take on any value

and discrete math topics like recursion

logic and community oryx help define the

basis of compute

signs what's your favorite topic and

Matt let me know in the comments section

and please subscribe for more

programming videos for now I've got to

check my logic so thanks for watching