Introduction to DM
Summary
TLDRThis video introduces discrete mathematics, contrasting it with continuous mathematics by focusing on countable, distinct objects. It emphasizes discrete math's role as a foundation for algorithms and logic in software development. The video explains how it enables logical differentiation, solving puzzles, creating complex passwords, and analyzing program execution times. It also touches on graph theory, allowing viewers to visualize and optimize networks. The presenter uses analogies, like a murder mystery, to illustrate the concept of mathematical proof, which involves logical deductions from assumptions to verify propositions, highlighting the importance of proof in establishing truth.
Takeaways
- 📚 Discrete Mathematics focuses on distinct, unique, and countable objects, such as the number of marbles in a jar.
- 💡 It serves as the language of logic, foundational for algorithms, and is crucial for software professionals.
- 🕵️♂️ Discrete Maths enables the differentiation of logical correctness in mathematical statements and arguments.
- 🔐 It is essential for creating diverse combinations, such as secure passwords, and has applications in hacking.
- ⏱️ Understanding discrete maths helps in analyzing the time an algorithm takes to execute, leading to better solutions and services.
- 🌐 Graph theory, a part of discrete maths, allows for the visualization of complex networks like social, communication, and transportation systems.
- 🛤️ With graph theory, one can determine the shortest route between two points, which is vital for logistics and navigation.
- 🔍 A mathematical proof is a method of verifying propositions through logical deductions from a set of assumptions.
- 🔑 Propositions are declarative statements that can be either true or false, forming the basis of logic and mathematical proofs.
- 🔄 The process of proving involves making a claim, establishing agreed-upon assumptions, taking logical steps, and drawing a conclusion.
Q & A
What is the primary distinction between discrete and continuous mathematics?
-Discrete mathematics deals with distinct, countable objects, while continuous mathematics focuses on objects that are continuous and cannot be easily counted, such as real numbers.
Why is discrete mathematics important for software professionals?
-Discrete mathematics forms the groundwork for logic, algorithms, and problem-solving in computing, which are essential for designing efficient programs, understanding data structures, and analyzing algorithms.
Can you provide a simple analogy for understanding discrete mathematics?
-Discrete mathematics can be seen as the 'superhero of counting and arranging things,' similar to how superheroes solve complex problems, discrete mathematics helps solve logical puzzles and optimize solutions in programming.
How does discrete mathematics help with solving logical puzzles?
-It provides the rules and structures, such as propositions and logical deductions, that allow one to break down problems and solve them systematically.
What is a proposition in the context of discrete mathematics?
-A proposition is a declarative statement that can either be true or false, and it forms the basis for logical deductions in proofs.
How is a mathematical proof constructed according to the script?
-A mathematical proof begins with a claim, followed by logical steps and evidence that support the claim, eventually leading to a conclusion that verifies the original proposition.
What example is given to explain the concept of proof?
-The speaker gives two examples: one involving a murder mystery where logical deductions lead to identifying the culprit, and another involving the sum of two even numbers, proving that the result is always even.
What are some practical applications of discrete mathematics mentioned in the script?
-Practical applications include creating strong passwords, analyzing the time complexity of algorithms, and solving problems in graph theory related to social networks, communication networks, and transportation.
What is the role of logic in discrete mathematics?
-Logic is fundamental in discrete mathematics as it provides the system for verifying propositions and deducing truths, which is critical in proofs, algorithms, and computational problem-solving.
What is the connection between graph theory and real-world networks mentioned in the script?
-Graph theory helps in visualizing and solving problems related to networks such as social, communication, and transportation networks by representing complex systems as graphs with vertices (nodes) and edges (connections).
Outlines
🧮 Introduction to Discrete Mathematics
The speaker introduces the topic of Discrete Mathematics (DM), contrasting it with continuous mathematics. DM deals with distinct, unique, and countable objects, exemplified by counting marbles in a jar. The speaker emphasizes the importance of DM in software development, likening it to a superhero's utility belt for counting and arranging. They outline the practical applications of DM, including logical reasoning, algorithm development, solving puzzles, creating complex passwords, and analyzing program execution times. The speaker also touches on graph theory, explaining how it can be used to visualize and understand complex networks and find the shortest routes between points.
🔍 Understanding Mathematical Proofs and Propositions
The speaker delves into the concept of mathematical proofs, defining them as a series of logical deductions that verify propositions. They break down the process of proving something, starting with a claim and building upon agreed-upon assumptions. The speaker uses a murder mystery analogy to illustrate how logical steps and evidence lead to a conclusion. They then provide a mathematical example, demonstrating how the sum of two even numbers is always even by using logical steps based on the assumption that even numbers are multiples of two. The speaker concludes by defining a proposition as a declarative statement that can be either true or false, setting the stage for further exploration of proof methods.
Mindmap
Keywords
💡Discrete Maths
💡Countable Objects
💡Logic
💡Algorithms
💡Proof
💡Propositions
💡Assumptions
💡Graph Theory
💡Passwords
💡Execution Time
💡Shortest Route
Highlights
Discrete maths is the study of distinct, countable objects, like counting marbles in a jar.
Discrete maths is referred to as the superhero of counting and arranging things.
It serves as a language of logic, foundational for algorithms in software development.
Discrete maths enables differentiation between logically correct and incorrect statements.
It allows for solving logical puzzles based on established rules.
The concept is fundamental in creating diverse combinations, such as for passwords.
It is a basis for understanding and potentially engaging in hacking activities.
Discrete maths helps in calculating the time an algorithm takes to execute.
Graph theory within discrete maths allows for visualizing complex networks.
It enables the determination of the shortest route between two points in a network.
A mathematical proof is the verification of propositions through logical deductions from assumptions.
Proofs are structured by starting with a claim, followed by logical steps, and concluding with a verification.
An example of proof is demonstrating that the sum of two even numbers is always even.
A proposition is a declarative statement that is either true or false.
Logic is a system based on propositions, with truth values being either true or false.
Highlights the importance of assumptions in the process of proving mathematical statements.
Logical deductions are compared to solving puzzles, where pieces are matched to form a framework.
The analogy of a murder case is used to explain the process of proving a claim with evidence.
Transcripts
hello folks welcome to this video on WE
defy discreete maths so here we'll start
with what is discrete maths mathematics
has two main branches one is discrete
and one is cous when I call it discrete
it means distinct unique
countable it is the study of distinct
and countable
objects okay for example imagine have a
jar of marbles you pick you pick one out
you see what is its color and you're
able to count how many such marbles were
there in the
jar so it is dealing with things you can
count that is the discrete
part yeah so you can think discrete
maths as a superhero of counting and
arranging
things as software professionals why do
I need discret Ms a bit subjective
details I'm going to give you it's a
language of logic it forms the
groundwork for algorithms and if you
take a film where you have a superhero
your discrete maths can be it is in
close analogy where you can call your
superhero pyic is your
discreet too subjective right so let's
see what you will be able to apply these
Concepts that you're going to learn in
this one thing is if you're given a
mathematical statement or argument
you'll be able to
differentiate okay if it is logically
correct or if it is in logical or
contains false informations all these
things you'll be you'll be able to
um
identify secondly you'll be able to
apply all these rules uh that this stre
month holds good for and based on these
rules you'll be able to solve logical
puzzles so see the word logic comes here
again and again and you'll be able to of
course create diverse combinations of
passwords what not diverse creations of
many many things and that is a basis for
hacking also if you interested in
it and uh there's one concept where
you'll be able to learn how you can
calculate U the time a program or a code
or a piece of code is taken to
execute given that when you know how
much time it's going to take for my
algorithm you'll be able to come up with
better Solutions you'll be able to
design better
Services right and uh you'll also be
able to analyze you'll be learning graph
Theory so you'll be able to visualize
everything as in a form of a graph right
so complex social networks communication
networks traffic networks whatnot
Transportation networks everything will
be a graph for you because you know the
concept behind a graph you know that it
is need of two Vex connecting on Ed
you'll be knowing all these things
finally you'll also be able to show me
here I know graph so this is the
shortest route I should take to reach a
place A to B call it I know I can I can
Define I can Define the exact and the
correct shortest route between two
dists so said that we will start from
proof what is a mathematical proof a
proof is the verification of
propositions by a chain of logical
deductions from a set of aumes oh that
was too much
right um we are natural problem solvers
carians are natural problem solvers so
let's try to
you know bring up an algorithm let's
break down big definition into small
small chunks and we take it from there
so proof is the verification of
propositions by a chain of logical
deductions from a set of aums there are
three important words if I get th get
them right I think I'm
sorted first one is Proposition second
one is logical connections or reductions
and uh set of aums third three words
let's take it slowly one by one but
before that what is a proof how do I say
how do I prove
something I usually if I want to prove
something I started with a claim yeah
I'm just going to say hey I think this
is true and let me show you why and as I
make my claim I'll take my first step
which is I'll say certain things and you
you guys will start agreeing okay yes
that seems to be true following that
following all your opinions following
that knowing that my assumption people
agree to it I take some logical steps
like jotting the puzzles one one piece
match it together form a framework one
one piece match it together bigger frame
finally I
conclude let's let's try to understand
this even more better I'll give you a
real word analogy Suppose there are two
people A and B and unfortunately um B
has been murdered
uh but I saw that A and B were sharing
the same room last night and I also
noted that uh there was a dagger in A's
hand with which B was killed so now I'm
going to start with a claim and my claim
is hey a only murdered
B and when I give you the claim no one
is going to be me but then I'm going to
start saying few things which all of you
will agree on so I say hey these two
people were only there last night
uh and uh a is only the last person that
b saw both of them were in the same room
and next day morning I found the dagger
in his hand and B is dead so when I say
this you come to an idea saying that oh
yes might be it could be true whatever
she say could be true then what I do is
I take some logic steps I need to prove
something right I cannot just accuse
someone based on my assumptions so I
take some logical steps I find the
fingerprints in the dagger it matches
with a I find a CCTV footage which
captures that a is attacking B this is
my logical steps that I need to take if
I take I'll finally conclude saying that
hey based on all this you know that my
IDE is
holding okay this is how you prove
something a mathematical
way um let's get a even more better
example I think this murder example is
bit absur so let's again start with a
claim
this time my claim is I'm going to say
sum of two even numbers is always even
I'm going to be even more precise I'm
going to say okayy the sum of six and
four is
even so when I say this I should say
some assumptions which all of you will
agree upon correct it's like bring two
three assumptions together and two
people will support it two three
assumptions together three people will
support it I'm forming a nego block
something kind
of my assumption here is I'm telling you
okayy guys listen to me even number can
be written as 2 multip by some
integer yes or my assumption is any even
number if you call it an even number it
can be written as two multiplied by some
integer an even number is a multiple of
two if you take four it is expressed as
2 multiplied by two if you take 10 it is
Multiplied as 2 multiplied 5 this is my
assumption when you hear my assumption
yeah you say hey yeah I think that makes
sense your assumption is right
so now I have to take some small logic
logic steps and end up proving my idea
so what idea am I going to prove I'm
going to say the logic here is very
fairly simple I'm going to say is it
visible think
my my screen is getting hidden okay the
logic that I'm going to say here is the
sum of our two even numbers 4 and six
can also be expressed as 2 * 2 + 3 4 + 6
is going to be 10 10 is expressed as 2 *
by 5 hence it is an even number and
hence I'm proving that the sum of two
even numbers is always
even and yes clearly 2 multi by 2 + 3 is
clearly a
multiple so this is how you when you
want to prove something this is how you
do this is the basis of it we'll start
developing this proof there are
different methods to proof uh to proving
something uh we'll we'll catch up that
line
short now what is this proposition we
said we started with proof and in the
definition I said it is set of
propositions like going back to that
slide yeah I said it's the verification
of proposition but what is a
proposition so here we go for what is a
proposition logic we are talking too
many logic words logic LC logic logic
logic is a system which is based on
propositions and a proposition is a
declarative statement is a declaration
that can either be true or false I can
see a declaration I can say okay this it
if I'm going to give you a sentence a
declarative sentence it can either be
true or
false okay that is called a
proposition we say the truth value of
the proposition is either true or false
if I show you the examples you'll able
to understand it much better and here
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