here I have the example if I'm skilled

at poker then I'm going to win a bunch

of

money so go to the casino I won a bunch

of money playing poker and I'm tempted

to conclude that therefore I'm skilled

at

poker okay so lots of people might

actually believe this this would be a

very reasonable fallacy that that a lot

of people would make let's try to

analyze it and I claim this is not true

and the loose idea is that just cuz I

went in one a bunch of money I might

have just gotten lucky I might be

terrible at poker but I got really lucky

so I cannot claim that I am skilled at

poker from this

argument so let's try to analyze the

structure of this argument first I have

if I'm skilled at poker I'm going to

give that a p then I will win money I'm

going to give that as a

q so that was my first premise if P then

Q so I'll write this as P implies Q q

that is my first premise next up it say

I won money playing poker so this is AER

q and then what we're saying is p

implies q and I'm going to I going to

copy it over here so we have it all in

place P implies Q then you assume the

conclusion and you're going to get I

skill that I'm skilled at poker so the L

The Logical form that you have here is p

implies Q then then q and then you get

out P but that wasn't modus pents that

wasn't modus tones it was none of these

and the problem here is that you're kind

of arguing by the converse it it's it's

Q implies P would be the converse of the

statement not P implies Q if you had Q

implies P here then you could take q and

get P but not the other way around so

this is trying to use the converse of

the statement and we know that a

Converse is not logically equivalent to

the original statement so this argument

does not work so this argument does not

work and we are going to say that it is

invalid I disagree with this logical

structure

Analyzing an argument for validity

Learning Objectives:
1) Analyze an argument to determine it's logical structure
2) Decide if the logical form of an argument is valid or invalid

The main argument forms are Modus Ponens, Modus Tollens, Generalization, Specialization, and Contradiction. To see more on each of these, check out the following playlist:

►Full DISCRETE MATH Course Playlist: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS

Other Course Playlists:

►CALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m

►CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n

►CALCULUS III: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd

►LINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6

► Want to learn math effectively? Check out my "Learning Math" Series: https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw

►Want some cool math? Check out my "Cool Math" Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho

*****************************************************
YOUR TURN! Learning math requires more than just watching math videos, so make sure you reflect, ask questions, and do lots of practice problems!

****************************************************

►Follow me on Twitter: http://twitter.com/treforbazett

BECOME A MEMBER:
►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join

MATH BOOKS & MERCH I LOVE:
► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett


let me do a really simple example next I

want to do the limit of a constant as X

goes to a value of a so for example I'm

going to bring up a graph here this is

the choice where my constant was one

what is the limit of one as X goes to a

well it doesn't matter where I am on

here if I if I want to talk about this

point right here the limit as you get

close to this point is one limit as you

get close to this point is one this

point every point you could get close

two the height of it's always one so no

matter what it is it's the value of one

or in the generic case when arbitrary C

here I'm going to say the limit is equal

to C okay here's sort of the next most

complicated function not a constant this

time it's the variable X so this is the

graph of a straight line it looks a

little bit like this and notice if I was

to do the limit as we get close to.5

here 0.5 is just equal to well 0. five

if I got close to the value of one up

here well that's going to have this

height of one over here if I go up to

two it's going to have this height of

two so as we get close to any point a

because it's just this straight line

that the height of it is going to be a

as and so the limit of the function X as

the variable X goes to a is just going

to be equal to a okay what about all

this nonsense we've got a whole polom

inside of here what we get to do is we

get to use these different rules these

limit laws to very quickly figure this

out and I'm going to go slowly this

first time but in a moment we're just

going to answer this

immediately the idea is first see was a

sum of three different things and we saw

that the limit of a sum was the sum of

the limits so I can break that up as the

sum of three different things the sum of

this first 3x cubed the sum of the X and

the sum of the minus one it's the it's

three different limits and we figured

out what the limit of a constant was and

the limit of x was we know both of those

and I'm also going to try to break this

one up notice we got a scalar and we got

a product X cubed is like x * x * X so

then when I Advance this what we're

going to have is first of all the the

triple product x * x * X is three

different limits here the scaler gets bu

built out of the front that was one of

our rules and then for the limit of x

and limit of the minus one I just plug

in those values we just saw the A and

the minus

one and now I've taken the X Cub down to

a limit of x X we just seen what that

was you plug in a and so what we're

going to get is 3 a cub + the Aus 1 in

other words what we've done here is we

just took this a that was there and we

just just plugged it in 3x Cub + x - 1

became 3 a cub + Aus one so polinomial

are really easy there's nothing to do

with a polinomial not have to graph it

you just plug the value in and indeed

it's true even slightly more General

it's true for something called rational

function so this is a rational function

f ofx it's a polinomial on the top and a

polom on the bottom and as long as you

don't have the polinomial at this point

a being equal to zero because you'd have

some sort of division by zero thing the

limit is super easy you just plug the

value in whatever rational function you

might have its limit is you just take

that a that you're taking the limit to

and you plug it in the same way this

polinomial we just plugged it in so for

a ton of these values as long as you

don't have a zero on the bottom as long

as you don't have that problem spot that

for these rational functions limits are

really easy you just plug them in

Building up to computing limits of rational functions

Watch first: Limit Laws: https://www.youtube.com/watch?v=dY5T7BcQ2Nc

We start with simple functions ( f(x)= c and f(x) = x ), and apply the Limit Laws of the previous video to build up to compute the limit of a class of more complicated functions: rational functions with nonzero denominators.

Learning Objectives:
1) Use graphs to compute the limits of "simple" functions
2) Apply Limit Laws in success to compute the limit of more complicated functions.
3) Apply the theorem about rational functions to quickly compute the limit of a rational function at points where the denominator is nonzero.

Now it's your turn:
1) Summarize the big idea of this video in your own words
2) Write down anything you are unsure about to think about later
3) What questions for the future do you have? Where are we going with this content?
4) Can you come up with your own sample test problem on this material? Solve it!

Learning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master calculus means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK.

This video is part of a Calculus course taught by Dr. Trefor Bazett at the University of Cincinnati.

BECOME A MEMBER:
►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join

MATH BOOKS & MERCH I LOVE:
► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett


let's look at an example to understand

what the composition of relations is in

this diagram we have two relations the

relation R from a set a to a set b and a

relation s from the set B to the set C

here are those two relations written out

precisely as sets for example we can see

under the relation r one is related to T

given this arrow that shouldn't be

surprising in the relation s for example

we see that s relates to Beta which is

indicated by this Arrow here s is also

related to Alpha and in the relation R

we can see that one is also related to

lowercase R so hopefully the diagram and

the set notation are both clear what

would happen if we compose these

relations what would it mean to compose

the relation r with the relation s it

works out probably just how you expect

under the composite relation for example

three relates to gamma because the

relation R relates 3 to q and then s

relates Q to gamma so in total under the

composition 3 is related to gamma in

plain language the elements relate from

a to c if there's a connection between

them given these two relations here is

the entire composite relation written

out you can see how I've notated the

composition we'll talk about notation in

a second but just for another example

you can see that 4 is related to Alpha

and 4 is related to Beta because 4 is

related to S but then s is related to

both Alpha and to beta one is related to

R but R isn't related to anything in C

so one only appears in the composition

once given that one relates to T and

then T relates to gamma so that's the

only time that 1 a p here is in the

composite relation now let's take a

quick look at the general definition of

a composition of relations you can see

how an ordered pair x z is in the

composition of relations if there's some

middle element connecting x and z you

probably notice the notation I'm using

here is the same as what's used for

function composition because of that

sometimes a different notation is used

since this is a composition of relations

not functions this semicolon notation is

one such option you may also recall that

with function composition the second

function is typically applied first the

order is from right to left there

doesn't seem to be an agreed-upon

standard for the composition of

relations in this case I'm going

opposite the way that it's done for

function composition I'm going left to

right the relation R relates X to y I

but then s relates y to Z and so in

total the composition relates X to Z

this is a left to right direction just

be wary you might also see a composition

of relations following the typical order

for function composition which is right

to left and that's a quick overview of

what a composition of relations Is let

me know in the comments if you have any

questions and thank you very much for

watching

[Music]

existence

[Music]

Composition of Relations | Discrete Math

We go over how to compose relations. The composition of relations is similar to the composition of functions, but in our example we'll see some differences in the notation for relation composition and all that good stuff. #discretemathematics

Intro to Relations: https://youtu.be/HXFHVRS1ZW8

Discrete Math playlist: https://www.youtube.com/playlist?list=PLztBpqftvzxXgpx57CEhZScNk2lWs1LdU
Discrete Math exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxW14CkPnOTIuGf2hUeX8jX0

◉Textbooks I Like◉
Graph Theory: https://amzn.to/3JHQtZj
Real Analysis: https://amzn.to/3CMdgjI
Abstract Algebra: https://amzn.to/3IjoZaO
Linear Algebra: https://amzn.to/43xAWEz
Calculus: https://amzn.to/3PieD1M
Proofs and Set Theory: https://amzn.to/367VBXP (available for free online)
Statistics: https://amzn.to/3tsaEER
Discrete Math: https://amzn.to/3qfhoUn
Number Theory: https://amzn.to/3JqpOQd

★DONATE★
◆ Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits: https://www.patreon.com/join/wrathofmathlessons
◆ Donate on PayPal: https://www.paypal.me/wrathofmath

Thanks to Loke Tan, Raül Beienheimer, Matt Venia, Micheline, Doug Walker, Odd Hultberg, Marc, Shlome Ashkenazi, Barbora Sharrock, Mohamad Nossier, Rolf Waefler, Shadow Master, and James Mead for their generous support on Patreon!

Outro music is mine. You cannot find it anywhere, for now.

Follow Wrath of Math on...
● Instagram: https://www.instagram.com/wrathofmathedu
● Facebook: https://www.facebook.com/WrathofMath
● Twitter:  https://twitter.com/wrathofmathedu

My Math Rap channel: https://www.youtube.com/channel/UCQ2UBhg5nwWCL2aPC7_IpDQ/featured


the next concept we want to introduce is

that of logical equivalence between two

different

statements now the idea here is that you

could have two statements and both of

them could be really long messy compound

statements of got lots of ores and ands

and Knots and all sorts of different

inputs and it's not clear whether those

two statements are the same thing but

just presented differently or whether

they truly are different things and the

way we're going to claim that two

statements that maybe superficially do

not look to be the same are in fact

logically equivalent if you have one you

get the other we're going to say they're

logically equivalent if their truth

tables are identical which is to say

we're saying something is logically

independent or logically equivalent

rather if it's the case that for every

pairing of their inputs all true false

pairings of their PS and their Q's that

their output is going to have the exact

same truth and falseness for every

different possible input so that is our

idea is having the same truth table so

here's a simple example I want you to

look at the statement not not p and and

we know that this idea of a double

negative is that a double negative in

English sort of cancels right if I say

that I am not not hungry then that's me

saying that I'm hungry that if I put a a

double negative in you can sort of

imagine it cancel it now let's just try

to verify that that this actually makes

logical sense according to what I've

told you which is the truth tables so

first of all we're going to go in and

we're going to try to fill in What's

Happening Here for p my two different

inputs are true and false so I always do

that first I I put all of the possible

input values that I could have my truth

or falseness of that input P so then if

I look at not P which is just sort of an

intermediate step here if p is true then

not p is going to be false and if p is

false then not p is going to be true so

I was just doing that as a temporary

thing now for the third column now I'm

looking at not not P so I'm going to not

p and taking the opposite of that so if

if not p is false then then the negation

of being false is true and so I'm going

to put a true over here and likewise if

my not p is true then the negation of a

true is false so I'm going to put a

false down here and then what I've

noticed here is that these two things

are going to be the same thing that the

the truth table for p which is a true

and a false is the same thing as the

truth table for not not P again a true

or false so this recovers for us the

idea that P is the same thing as not not

p and the way that I like to write this

I have my statement P that's my my one

and I'm trying to compare it to

something else and I use if it's logical

equivalence I use an equality sign but

one that's got like three horizontal

lines to it that's what I mean for

logical equivalence that's going to be

my short hand for it this is logically

equivalent to not not

p and it's going to turn out that

there's all sorts of different logical

forms that may appear to be very

different like not not P superficially

is in a different way than P but even

though it's superficially different they

truly are the same thing and our method

is just write out the truth tables for

both of them are they the same truth

table or are they different

Logical Equivalence of Two Statements

Two statements are Logically Equivalent if they have the same truth table. For instance, p and ~(~p) are logically equivalent.

****************************************************
YOUR TURN! Learning math requires more than just watching videos, so make sure you reflect, ask questions (leave them in the comments!), and do lots of practice problems.
****************************************************

►Full Course Playlist: DISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS

****************************************************
Other Course Playlists:

►CALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m

►CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n

►LINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6

***************************************************

► Want to learn math effectively? Check out my "Learning Math" Series:
https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw

►Want some cool math? Check out my "Cool Math" Series:
https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho

*****************************************************

►Check out my 2nd Channel for lower production quality "live" math videos: https://www.youtube.com/channel/UCFNL8G1CZ4dH0p-Jf7UXqdQ

*****************************************************

►Follow me on Twitter: http://twitter.com/treforbazett

*****************************************************

This video was created by Dr. Trefor Bazett


we've had a similar goal for quite some

time

it was compute the derivative other as

many functions as possible and we have

done a lot we figured out polynomials

and trigonometric terms we figured out

how to do things like the product we

figured out how to do things like a

composition we know how to do quotients

we know how to do sums we know an

enormous amount but there's one class

that we don't know yet and that's

Exponential's for example we do not know

how to take the derivative yet of X to

the power of sine of X so how can we

solve something like this what I want to

use what I'm going to exploit is a nice

little fact about logarithms and the

fact is this if I'm going to take the

natural log of a e to the power of B

then one of the rules of logarithms is

that this exponent that I have it can

come brought down at the front B times a

natural log of a now I want to suppose

you're taking the derivative of this

if exponents are hard things like X to

the sine X if those are hard well we do

know what the derivative of a product is

we've got the product rule for that

so because the natural log sort of

transforms exponents into products it's

going to be able to transform something

that's challenging into something as

effectively just a product rule so let's

see how this actually works in an

example I take my y equal to X the power

of sine of X and before I just try to

compute dy DX I'm going to go and take

the natural log of both sides first that

is I'm going to compute the natural log

of Y and the natural log of the X to the

power of sine of X and then our goal is

going to be if I have it written in this

form I'm gonna take the derivative of

both sides here now why have I helped

myself the key idea is this I've got the

X to the sine of X this is kind of like

an a to the B and I can apply my log

rule I can take this sine of X and I can

bring it down so what this is gonna have

on the right hand side is the derivative

of the sine of X times the natural log

of X what happened over here on the left

well this is a chain rule an outside

function the natural logarithm the

inside function y of X so what's its

derivative

it's the drew the upside which is one

over whatever the inside is one over Y

times the derivative the inside dy/dx

that's me apply my chain rule so apply

my chain rule over here I applied a log

rule over there I can take the

derivative of product another take the

derivative of sine times the natural log

of X I just use the product rule so what

do I have I'm going to leave the left

hand side sine goes to coast plus the

natural log plus sine times the dru the

natural log which is 1 over X so I've

applied my product rule here ok so I'm

happy with this but I want to do a

little bit more work let's get rid of

all this other nonsense what I want to

do is I don't want this Y here any

longer

I want my derivative to be some function

of X so I want to manipulate this but I

can both move the Y up to the other side

and I know that Y is X to the power of

sine of X so I can isolate it and take

this Y here move it over and what I get

is the derivative ektu X is there's the

Y exit sine of x times all of this other

stuff so that is my final answer for the

derivative with respect to X of this

function X to the sine of X and you can

see that it is kind of a long and messy

expression here but this is just how

this logarithmic differentiation works

it makes it possible for us to compute a

derivative that we didn't know how to do

before

Logarithmic Differentiation |  Example: x^sinx

Description:
One of the few major classes of functions we can't yet differentiate are functions like x^sin(x). That is, one function to the power of another. By taking ln(x) first we can use the power of log rules to convert this into a product, and then apply product rule. But we have to be careful to convert back to the original function at the end!

Learning Objectives:
1) Compute the derivative of one function to the power of another.

Now it's your turn:
1) Summarize the big idea of this video in your own words
2) Write down anything you are unsure about to think about later
3) What questions for the future do you have? Where are we going with this content?
4) Can you come up with your own sample test problem on this material? Solve it!

Learning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master calculus means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK.

This video is part of a Calculus course taught by Dr. Trefor Bazett at the University of Cincinnati.

BECOME A MEMBER:
►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join

MATH BOOKS & MERCH I LOVE:
► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett


if you recall when we talked about a

normal implication P implies Q we've

seen before that P implies Q is

logically equivalent to not Q implies

not P so I'm going to make use of that

logical equivalency when I try to go out

and prove things in particular my first

goal is going to be to try to prove an

implication that's what I'm trying to do

I'm trying to show prove an assumption

leads to a conclusion but what I'm going

to do is treat this as the other way

around I'm know that P implies Q is the

same thing as not Q implies not P that

is I'm instead going to go and try to

prove the Contra positive I'm going to

try to prove not Q implies not p and

that's the way I'm going to demonstrate

that P implies Q so let's see an

example in this theorem I'm saying if

the square is even then n itself is also

going to be even now I want to not

carefully this is different from the

theorem that we've seen in a previous

video this is not not saying if n is

even then n s is even it's the other way

around it's saying if n squar is even

then n is going to be even I'm going to

just try to use a direct proof and show

you why it might be challenging so my

assumption might be that N2 is even

which is going to be that N2 is going to

be twice some k for some K I'm not going

to write all my symbols down because

this is my plain around step I don't

have to be all that

precise all right and I want to say

something about n I've got this Square

here so maybe what I should be doing is

taking a square root of both sides that

might give me my n if I took the

positive square root for instance but

then what I have is it being like equal

to < tk2 * < TK K which doesn't really

look anything at all like being even so

assuming that my n squar was even and

taking square roots just didn't leave

anywhere it's got all this messy rot2

stuff floating around so proving it

directly seems a little bit challenging

so let's let's get rid of

that since proving this directly wasn't

all that useful let's try to prove it by

the contrapositive we're we're going to

do it the other way around with

negations so when I write it as proof by

contradiction by Contra positive rather

note what happens first of all the the N

squar is even and the N is even they

flip orders but they also get two knots

in that's what happens when I do Contra

positive so in other words I'm assuming

that n is not even and I'm trying to

deduce that n^2 is not even by the way

the Assumption n is not even is the same

thing as n is odd and the conclusion n

squ is not even is the same as the

conclusion n s is odd but I've actually

seen that particular proof before I've

seen a proof that n is odd implies N

squared is odd and so we can sort of

bring that one in and this is the proof

that we've seen before

indeed this claim n^2 even implies n is

even is the same claim it's logically

equivalent to n is odd implies n squ is

odd it's just that when it's written

this way in its contrapositive form it's

quite tractable and we were able to do

its proof almost directly it sort of

just fell out by our assumptions in a

way that just wasn't true when I tried

to directly prove the original so it's a

wonderful illustration of how a a proof

that's challenging and hard if you try

to go and prove its condra positive the

condra positive while logically

equivalent might be much easier to prove

Proof by Contrapositive | Method & First Example

Learning objective: prove an implication by showing the contrapositive is true.

This video is part of a Discrete Math course taught at the University of Cincinnati.

BECOME A MEMBER:
►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join

MATH BOOKS & MERCH I LOVE:
► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett


[Music]

so let's look at an example of

computing the mass using line integrals

so we have the following example which

is a wire is in the shape of a circle of

radius 2

centered in the origin let's say and has

density function

rho which is 2 plus 1 half x

we want to find the mass of the wire now

of course the mass is given by the line

integral over the curve c

of the density function and so we first

have to parameterize the wire

and we'll do that by using the function

alpha t which is

2 sine t in the x component and 2

cosine t in the y component and we'll

vary t

from minus pi to pi now with respect to

this parametrization we see

that rho is just two plus sine t

in particular the line integral over the

curve c of rho of x

y d s is just the integral from minus pi

to pi

of two plus sine t that's just rho with

respect to the parametrization

and then the arc length term is just the

square root of 4

cosine squared t plus 4 sine squared t

dt the pythagorean identity of course

tells us that cosine

squared plus sine squared t is one so

we're left with the integral from minus

pi to pi

of two times two plus sine t dt

which is just two times two t minus

cosine t

from minus pi to pi and this expands to

two times two pi minus cosine of pi

minus two times minus pi plus cosine of

minus pi

of course since cosine of pi is minus

one and cosine of minus pi is also

minus 1. what we're left with is 2 times

pi plus 1 plus 2 pi minus 1

and this is just 8 pi

you

Line Integrals, Mass, and Density | Vector Calculus | Calculus 3

Vector Calculus and Differential Equations Playlist -- https://www.youtube.com/watch?v=UhuxxkOocdY&list=PL912tg7wFUfXIwSKeTyxU5LTqdRZ8Urza&index=1

More videos on geometric algebra and the exterior algebra:
What is area? -- https://www.youtube.com/watch?v=3sGNR36eUR0
Why you don't understand the fundamental theorem of calculus -- https://www.youtube.com/watch?v=JV0vgVa4bCg
Why you don't understand Green's theorem -- https://www.youtube.com/watch?v=XEDtm09ol5o
Computing gradient vector fields -- https://www.youtube.com/watch?v=oNKFhQTn-1s
Line Integral Example 1: https://www.youtube.com/watch?v=fmnYFM0C9W8&ab_channel=KyleBroder
Line Integral Example 2: https://youtu.be/dW7SdW5sRuY
Line Integral Example 3: https://youtu.be/FXHYaFig-OE
Older video on Line Integrals, Mass, and Density: https://youtu.be/sYmSGrcA10k
Stokes' theorem and Path Independence: https://youtu.be/2FGKxFG5lSI
Conservative Vector Fields: https://youtu.be/eQhj4m8pt9E
Computing Arc Length: https://youtu.be/UhuxxkOocdY
Introduction to Vector Calculus: https://youtu.be/fV0d-FE03EM
How conservative vector fields relate to Euler's number: https://youtu.be/oU5elvZL0uU

👍 To support the channel, hit the like button and subscribe.

🐸 To learn how I make these animations using Grant Sanderson's (3Blue1Brown) program Manim, check out my quick intro to Manim on Skillshare: https://www.skillshare.com/r/profile/Kyle-B/344454857

🎸 Music in this video is from Epidemic Sound: https://www.epidemicsound.com/referral/yi0wzw/
Ways to support the channel:
🏆 Support the channel over on Patreon https://www.patreon.com/KyleBroder
🐸 Support the channel by signing up to a free trial of Skillshare using the link https://www.skillshare.com/r/profile/Kyle-B/344454857

FYI, these are affiliate links 💪🙏

These videos are separate from my research and teaching roles at the Australian National University, University of Sydney, and Beijing University.

Hi, my name is Kyle and I'm currently doing my doctoral mathematics degree in complex differential geometry under the supervision of Professor Gang Tian and Professor Ben Andrews.


the spookiest math curve is the infamous

Witch of anessi originally named by

Maria anessi the first woman appointed

to the math professor it was then later

completely mistranslated to woman who is

against God and then to which now we can

construct it geometrically by starting

with the circle of radius a then taking

a point on the circle and drawing the

segment from the origin out to the point

P if you put a horizontal tangent at the

top of the circle and extend our segment

out until it intersects with that

horizontal finally drop it de down until

it matches the height of the point P

this gives us a new Point as we move

around the circle you sketch out a curve

this is the witch of anessi analytically

this curve is given by 8 a cub / x^2 + 4

a^ 2 where a is the radius of the circle

and I'm a particular fan of the special

case where a is equal to 12 and in this

case it simplifies to just be 1x^2 + 1

now any good mathemagician can make Pi

appear so let's integrate from 0 to 1

and the calculus students among you will

recognize that this is the derivative of

arctan and so evaluating at 0 and 1

gives the value of pi over 4 but if

instead we had looked at the inside and

expanded using geometric series to give

1 - x^2 + X4 - x 6 and so on then

integrate term by term evaluated at the

zero and the one and you get the series

1 - 1/3 plus 5th - 17th and so forth

this is a series expansion for pi

Spookiest Math Curve: Witch of Agnesi

Spooky!!
BECOME A MEMBER:
►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join

MATH BOOKS I LOVE (affilliate link):
► https://www.amazon.com/shop/treforbazett

COURSE PLAYLISTS:
►DISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS
►LINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6
►CALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m
► CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n
►MULTIVARIABLE CALCULUS (Calc III): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd
►VECTOR CALCULUS (Calc IV) https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa
►DIFFERENTIAL EQUATIONS: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxde-SlgmWlCmNHroIWtujBw
►LAPLACE TRANSFORM: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxcJXnLr08cyNaup4RDsbAl1
►GAME THEORY:  https://www.youtube.com/playlist?list=PLHXZ9OQGMqxdzD8KpTHz6_gsw9pPxRFlX

OTHER PLAYLISTS:
► Learning Math Series
https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw
►Cool Math Series:
https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho

SOCIALS:
►X/Twitter: http://X.com/treforbazett
►TikTok: http://tiktok.com/@drtrefor
►Instagram (photography based): http://instagram.com/treforphotography


the final example we're going to do is a

incredibly important example of a vector

field one

that comes from just a normal old scalar

function f

of x y and z a a function from r

three in this case just to r well for

such a scalar function

you could take what we called previously

in multi-variable calculus the gradient

the i-hat component is the partial

respect to x the y-hat is the partial

with respect to y

and the k-hat is the partial with

respect to z

that is if you give me a scalar function

f from either r2

or r3 to r then i can give you a vector

field a corresponding vector field given

by the

gradient of f the gradient of f is a

vector field

and it also has a very nice geometric

meaning

so for example suppose that my function

was the function f of x y

equal to x squared plus y squared then

we could compute out what its gradient

is not too bad well the partial respect

to x is just 2x and the partial respect

to y

is just 2y so the gradient is just that

now we've seen this vector field except

for the fact that there's a 2 in both

places which just stretches every vector

by a multiple of 2.

this is precisely the vector field we

saw earlier

and so now i can put everything together

in one picture so this is my

picture for this function what we see is

the graph of the function that's this

blue paraboloid

and then in the domain the domain is the

x

y plane we get a vector field in that

domain

and so i plotted the gradient vector

field in the domain

and then this function above the domain

and

as you might recall from multi-variable

calculus

the gradient vector points in the

direction of steepest

increase of your function so if you're

at some particular x and y-coordinate

and you're up there on the function

value at that x-y coordinate you want to

know

how do i climb this mountain represented

by this function as fast as possible

where is this function steepest what

direction should i go you should go in

the direction of the gradient

i hope you enjoyed this video and if you

did please give it a like for the

youtube algorithm because we're all

mathematicians here and we appreciate

algorithm just as much as youtube does

if you have a question leave it down in

the comments and

we'll do some more math as always in the

next video

The Gradient Vector Field

One prominent example of a vector field is the Gradient Vector Field. Given any scalar, multivariable function f: R^n\to R, we can get a corresponding vector field that has a precise geometrical meaning: the vectors point in the direction of maximal increase of the function.

MY VECTOR CALCULUS PLAYLIST:
►VECTOR CALCULUS (Calc IV) https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa

OTHER COURSE PLAYLISTS:
►DISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS
►LINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6
►CALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m
► CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n
►MULTIVARIABLE CALCULUS (Calc III): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd
►DIFFERENTIAL EQUATIONS: https://www.youtube.com/watch?v=xeeM3TT4Zgg&list=PLHXZ9OQGMqxcJXnLr08cyNaup4RDsbAl1

OTHER PLAYLISTS:
► Learning Math Series
https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw
►Cool Math Series:
https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho

BECOME A MEMBER:
►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join

MATH BOOKS & MERCH I LOVE:
► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett

SOCIALS:
►Twitter (math based): http://twitter.com/treforbazett
►Instagram (photography based): http://instagram.com/treforphotography


well it's true that generally and

implication and it's converse are not

the same thing they don't have the same

truth value what you can have is a

scenario where both directions this the

original conditional and it's converse

where those are both true and when both

of those are true what we can have is

something called the biconditional

and you'll notice here that in the

biconditional it's got this sort of

double sided arrow because it's saying

that P implies Q and it's the case that

Q implies P this is the idea where there

P and Q are sort of like equivalent

property if you have one you get the

other and if you have the other you get

the initial one so let's try to figure

out what this works in terms of some

statements if I study hard then I will

pass so that was the statement P implies

Q but I can also go the other way around

this probably applies to most of us

I could say if I pass then I studied

hard so this is the statement Q implies

P and you'll notice that what I'm having

here is if I combine these together and

I think of them as one statement

it's a conjunctive statement staying on

that and in it it got one implication P

implies Q and it got a second

implication Q implies P and the way you

should think about this is that if we

exclude people like me or people who

have a lot of amelie arity with

materials they can sort of sit down and

not bother studying at all and just pass

the test already

although truly for myself that's just

good I studied a long time in the past

if we exclude those kinds of cases this

is probably true you need to study hard

in order to pass and if you pass you did

indeed study hard so both of these

different implications are going to be

true and I can combine them in a kind of

nice way look at my free zoology here I

say I will pass I use this funny word

funny phrase if and only if I study hard

so whenever we write that whenever we

say they sort of guess and only a thing

here this is a way of saying that both

directions are going to be

it's a way of saying if I study hard

then I will pass and if I passed then I

study hard that's what I have here but

the first of these ifs is referring to

the stop statement and the second of the

one that said only it is referring to

the other way around if I pass the neck

started hard so mathematician is really

like this phrase if and only if as a

sort of a shorthand for talking about by

conditionals as a shorthand for having a

conjunctive statement with two different

conditionals written down they just put

if one thing this is the passing the

thing on what it has that's going to be

true if and only if I study hard

Biconditional Statements | "if and only if"

A biconditional statement is one of the form "if and only if", sometimes written as "iff". The statement "p if and only if q" means "p implies q" AND "q implies p". That is, it is a conjunction of two individual conditional statements. In this video we will see how to write and interpret sentences using if and only if notation.

►Full DISCRETE MATH Course Playlist: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS

Other Course Playlists:

►CALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m

►CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n

►CALCULUS III: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd

►LINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6

► Want to learn math effectively? Check out my "Learning Math" Series: https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw

►Want some cool math? Check out my "Cool Math" Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho

*****************************************************
YOUR TURN! Learning math requires more than just watching math videos, so make sure you reflect, ask questions, and do lots of practice problems!

****************************************************

►Follow me on Twitter: http://twitter.com/treforbazett


now I want to look at an example where

this vacuously true part applies

vacuously true meant that when the

hypothesis was false so it didn't even

get off the ground then we were going to

still say that the true the statement is

true but but true in a sort of

unimportant or or uninteresting or

vacuous sense so consider this statement

if Trevor is a unicorn then everyone

gets an A in the class so it is set up

this this is an if P then Q I can say

this is an if

P then Q statement or alternatively I

can say it is a p implies Q

statement now what we know is that

because I can turn a conditional like

this into a disjunctive into an or

statement of the form not P or Q let me

try to rewrite this in that form so in

this statement what I've said is either

Trevor is not unicorn so this is the

statement not P

now or everyone gets an a that's the

same statement I had before that is

again

Q when it's written in this form however

when it's written in its disjunctive

form I think it's more obvious that we

should say that that is true either

Trevor is not a unicorn or everyone gets

an a one of those two things is true it

turns out to be unfortunately the uh

Trevor is not a unicorn one however we

still have this logical form that is is

equivalent to the first one that's

written in its conditional or

implication form and it's because of

examples like this that we say that it's

vacuously true yes this statement down

here is true uh one of these two things

in myor statement is true the Trevor is

not a unicorn but it's not interesting

that it's not an important conclusion

the statement doesn't really mean

anything I can say anything I want here

if pigs fly then you get a million bucks

uh all of these statements are sort of

true but true in a vacuous sets

Vacuously True Statements

Learning Objectives: Determine when a conditional statement is vacuously true

A conditional statement of the form "if P then Q" is called vacuously true whenever P is false. This might seem counterintuitive, but consider the reverse. The conditional "if P then Q" is false whenever the assumption P is true and the conclusion Q is false. So when the assumption is false it doesn't matter what the conclusion is, and we call the conditional true, but in a vacuous sense.

****************************************************
►Full Course Playlist: DISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS

****************************************************
Other Playlists:

►CALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m

►CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n

►LINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6

► Want to learn math effectively? Check out my "Learning Math" Series: https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw

►Want some cool math? Check out my "Cool Math" Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho

*****************************************************
YOUR TURN! Learning math requires more than just watching videos, so make sure you reflect, ask questions, and do lots of practice problems!
****************************************************

►Follow me on Twitter: http://twitter.com/treforbazett

This video was created by Dr. Trefor Bazett


so now that we understand what a

proposition is let's go ahead and

introduce some notation in particular

because we're writing out these

propositions and the full form like this

can get really tedious so we use some

notation so that notation is referred to

as propositional variables so we use

propositional variables these are also

known as so aka sentential variables and

we use these things to represent

propositions and we typically use the

symbols like p q r s etc to represent

these different propositions and these

variables right here

can I have to take the value of true

because remember a proposition can

either be true or false and so we

usually represent true using a tea right

here or false F so for example instead

of having to write down one plus one

equals two every time I want to refer to

that proposition right there I could

just say well I'm going to define that

to be the propositional variable P and

then this proposition right here I can

define that to be the propositional

variable Q and this longer proposition

right here I can very compactly

represent that simply by using the

propositional variable or sentential

variable R right here and as this

notation right here indicates these

things only take on the value of true or

false so for example if we were to look

at our different variables and also look

at their truth value then in this

particular example right here our prop

up proposition P turns out to be true

because yes indeed one plus one equals

two whereas Q over here that proposition

it takes on the value false right there

and our also happens to be true so these

are propositional variables that have

particular truth values this just gives

us a really compact way to refer to what

could be a very long declarative

sentence right here by the way this

thing right here is referred to as a

truth table and it's going to show up

quite a bit as we talk about

propositional logic

propositional variables


Hey! Can you cover complex analysis? [Chatters
about people requesting complex analysis over the years]

Alright let’s do this. But why
should anybody care about complex analysis?

Well, just like any other field of maths,
it is beautiful, which is enough for pure

mathematicians, but for applied mathematicians,
physicists, engineers and the like, there

are quite surprisingly lots of profound applications.
It is very useful in solving differential

equations, because the real and imaginary
parts of a complex function satisfy a really

important differential equation. This equation
arises in many areas of physics, like electrostatics,

fluid dynamics, and heat conduction, but even
if you are not solving this particular equation,

for any other DEs, a common trick is to use
Fourier or Laplace transforms, which requires

tools in complex analysis.

If that’s not tantalising enough, it is
connected to some popular math topics like

the fundamental theorem of algebra, and the
Riemann hypothesis. Well, if even that is

not attractive enough, it allows you to show
off! No, seriously, how would you evaluate

this integral? Using partial fractions would
be a nightmare, but if we consider its complex

version, integrating on a complex plane, it
becomes massively easier. And actually, these

are only applications that hopefully *most*
of you would find interesting, but obviously

there are more! All I’m trying to say is
that complex analysis is very powerful!

And so I am going to make a video series about
complex analysis, and this is my plan. The

first one is this video, a preview of what’s
coming next. Then a crash course on complex

numbers, so that we can all be on the same
page, and then how to visualise complex functions,

and the messy stuff about complex functions.
And then a particular kind of complex function,

the Möbius map, and the slick way of visualising
it using the Riemann sphere. We can’t not

talk about complex differentiation, but it
turns out that if we want to visualise integration,

we also have to know a bit of vector calculus
first. But after all these, we are going to

see *some* of the applications mentioned,
possibly in several videos.

However, I want to emphasise that it isn’t
like this hasn’t been seen anywhere on YouTube.

Welch Labs made a series on complex numbers
and functions, Möbius transformations were

illustrated more than 10 years ago in this
video, Zach Star made a video on complex derivatives,

loads of people have covered vector calculus
before, and actually, this whole series will

be very much from this amazing book called
“Visual Complex Analysis” by Tristan Needham.

So think of this series as a more convenient
compilation and *my* take and animation on

this subject. However, a little note that
might be disappointing to the purists. This

video focuses much more on visual insights,
and the rigour is not prioritised in the arguments.

By all means, make the arguments in the videos
more rigorous! But in any case, I hope you

will enjoy the series ahead!

Thanks to these patrons, and as always, like
and subscribe with notifications on, comment

and share the video as well! See you next
time!

Why care about complex analysis? | Essence of complex analysis #1

Complex analysis is an incredibly powerful tool used in many applications, specifically in solving differential equations (Laplace's and others via inverse Fourier / Laplace transforms), and of course, fundamental theorem of algebra, Riemann hypothesis, as well as solving complicated integrals to show off!

This is the start of a series on complex analysis, which focuses on the visual insights rather than the more traditional rigorous approach, so it is not very likely that we will touch upon more advanced, yet very remarkable theorems in complex analysis, like Riemann mapping theorem, or the Picard's theorems. This is more likely of more use to the applied mathematicians, physicists and engineers, at least in the first few videos, simply because the interesting theorems are most likely pushed towards the end, and grouped as "applications of complex analysis", like FTA and RH.

This is only currently my plan, and things are likely to change, but hopefully this is the backbone of what's coming next. I might not be uploading like weekly, but hopefully as frequently as I hope.

Videos mentioned:
https://www.youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF (Welch Labs complex numbers playlist)
https://www.youtube.com/watch?v=0z1fIsUNhO4 (Möbius transformations video)
https://www.youtube.com/watch?v=b8_3PFjiJvY (Zach Star complex derivatives)
https://www.youtube.com/playlist?list=PLSQl0a2vh4HC5feHa6Rc5c0wbRTx56nF7 (Khan academy playlist on vector calculus)
https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa (Dr Trefor Bazett’s playlist on vector calculus)

Some (sort of) unexpected applications / connections to complex analysis:
https://en.wikipedia.org/wiki/Joukowsky_transform (historical aerofoil application)
http://math.colorado.edu/~kstange/papers/notes-Spin.pdf (Relationship between Möbius map and special relativity)

Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6

If you want to know more interesting Mathematics, stay tuned for the next video!

SUBSCRIBE and see you in the next video!

If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!

Social media:

Facebook: https://www.facebook.com/mathemaniacyt
Instagram: https://www.instagram.com/_mathemaniac_/
Twitter: https://twitter.com/mathemaniacyt
Patreon: https://www.patreon.com/mathemaniac (support if you want to and can afford to!)

For my contact email, check my About page on a PC.

See you next time!


associated to every matrix there are two

canonical subspaces but a null space of

a column space and this video I want to

figure out what is the dimension of the

null space and what is the dimension of

the column space now we know that

dimension is just the number of basis

vectors and in fact in previous videos

we've actually seen how to figure out a

basis for the null space and a basis for

the column space so you just figure out

those bases and that's gonna be the

dimension so in this video I'm gonna

quickly go through that process and in

the next video we're going to see some

kind of crazy which is how the dimension

of the null space of how the dimension

of the column space actually relate to

each other and are interconnected so

first off dimension of the null space

I'll remind you that the null space is

all the vectors that are killed off by

applying particular matrix a they're all

the vectors where ax is equal to 0 and

my steps to find a basis for this are

number one I want to go and reduce the

matrix so but ax equal to zero I wanna

see maybe like some example like this

where I've got my Augmented majors with

the zero column I'm gonna put it

actually into our our EF form for you

then what I'm going to do as I say that

there's these two free columns are going

to give me variables for that I read off

of the rows they're gonna tell me what

the rows are going to do and I can put

it then into vector format now if you're

not really speedy with this pause the

video and do it yourself carefully but

the point I want to make is this when I

go through the standard process of

Gaussian elimination and back

substitution to put the vectors in

vector form what I get out of it is a

basis for the null space these two

vectors over here they're linearly

independent and indeed they're by

definition going to be spanning all the

vectors since every vector can be

written in this way so that's the basis

so in this case I've got a big sis it's

got two different vectors in it and

since the dimension is going to simply

be the number of basis vectors the

dimension of the null space is going to

be 2 here so my steps are number one to

reduce the matrix as we just saw number

two to note that the number of free

variables is the same thing as the

number of basis vectors and then we can

say that

dimension of the null space is just

going to be whatever the number of free

variables is going to be so that's

dimension of the null space what about

dimension of the column space now well

what's the column space remember it is

the span of the cause of matrix a and

indeed because the basis has to both

span and be literally independent the

only thing to check in the column space

is the move your independent part cuz it

gets a span part for free so if I take

my matrix like here's the same a I want

to reduce it in the exact same way

because here's the trick if I look at

the first and second column here well

there are multiples of each other

the second column is just twice the

first column so this if I look at all

four vectors they're not linearly

independent however if I focus just on

the vectors with leading ones that still

spans it and that are going to be

linearly independent so the color

leading ones are going to be my basis

for the column space and therefore the

dimension of the column space is just

going to be the number of leading what's

now in the next video what we're gonna

see is we've got the dimension of the

column space we've got the dimension of

the null space but in fact they are

intimately related

Computing Dimension of Null Space & Column Space

The dimension of a subspace is the number of basis vectors. For the two canonical subspaces associated to any matrix - the Null Space and the Column Space - we repeat quickly the computation of basis vectors for them and thus are able to compute their dimensions.

**************************************************
►Full Playlist: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6
**************************************************

**Now it's your turn**
1) Summarize the big idea of this video in your own words
2) Write down anything you are unsure about to think about later
3) What questions for the future do you have? Where are we going with this content?
4) Can you come up with your own sample test problem on this material? Solve it!

Learning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master math means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK.

***************************************************
Want more ideas for learning math effectively?
►How to Watch Math Videos:
https://www.youtube.com/watch?v=Ljq5y3ksk8o

►5 Tips to Make Math Practice Problems Effective: https://www.youtube.com/watch?v=LPH2lqis3D0

****************************************************
►Want some cool math? Check out my "Cool Math" Series:
https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho

****************************************************
Course Playlists:
►Calculus I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m

►Calculus II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxdQCBjYswqbn7LxL1pW4cW4

►Discrete Math: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS

►Linear Algebra: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6

*****************************************************
►Follow me on Twitter: http://twitter.com/treforbazett
*****************************************************

This video was created by Dr. Trefor Bazett, an Assistant Professor, Educator at the University of Cincinnati.

BECOME A MEMBER:
►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join

MATH BOOKS & MERCH I LOVE:
► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett


in this example we want to figure out

the probability that if I randomly

choose a number between 3 and 14 where

I'm including the end I'm cluding 3 and

I'm including 14 what is the probability

that an even integer now what I want to

partially illustrate here is that this

questionable probability is

intrinsically tied to the idea of

counting in that that in many ways

answering your probability problem is is

answering a question about counting so

let's pull up the particular integers

that were interested in so I've got 3 4

all the way down to 14 and I've

highlighted in yellow the even integers

so what our formula told us is that I

want to investigate the number of things

in our event so this case the number of

evens happens to be denoted by an E as

well but it stands for event divided out

by the number in our sample space and

that this is going to be the probability

of our event so we have to do these two

counties so let's count the even ones

first I've got 4 6 8 10 12 and 14 so

that looks like 6 even are appearing

here so this is going to give me 6 and

then the total number of things between

3 and 14 is 12 so divided up by 12 here

in other words this is going to be 0.5

or 50% as well but notice that if I

change this question a little bit

suppose I then went and said from all

the way from 3 up to 15 well how does

that change 15 is not an even integer

but it would increase the number down

here from 12 to 13 and so your

probability would change because you've

increased the size of your sample space

even though you haven't changed the

number of even if you haven't changed

the number in your event

Example: Computing Probabilities using P(E)=N(E)/N(S)

This video computes the probability of an event using the formula P(E)=N(E)/N(S). That is the probability of an Event occurring counts the number of times that event occurs an divides by the total number in the sample space.

►FULL DISCRETE MATH PLAYLIST: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS

OTHER COURSE PLAYLISTS:
►CALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m
► CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n
►MULTIVARIABLE CALCULUS (Calc III): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd
►VECTOR CALCULUS (Calc IV): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa
►DIFFERENTIAL EQUATIONS: https://www.youtube.com/watch?v=xeeM3TT4Zgg&list=PLHXZ9OQGMqxcJXnLr08cyNaup4RDsbAl1
►LINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6

OTHER PLAYLISTS:
► Learning Math Series
https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw
►Cool Math Series:
https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho

BECOME A MEMBER:
►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join

MATH BOOKS & MERCH I LOVE:
► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett

SOCIALS:
►Twitter (math based): http://twitter.com/treforbazett
►Instagram (photography based): http://instagram.com/treforphotography


recall that if I had two different

logical statements P and Q then I could

have the conditional between them that

said if P then Q and I can do the same

idea not just between two different

statements but between two different

predicates so that's what the idea of a

universal conditional is so we use this

notation P of X and notice that it's a a

double arrow opposed to a single arrow

here a P of X which is a predicate is

going to imply a different predicate Q

of X and what I mean by that is that

every time I put it in an input then the

predicate becomes a statement so what

I'm saying is for every possible input

in my domain or for every X inside of my

domain then the statement P of X where I

plugged in a specific value of X implies

the statement Q of X for that same value

of x so in other words a universal

conditional is going to be just like a

normal conditional but where your

statements are replaced by predicates

and then the condition will work for

every single input value so let's take a

look at this mathematical sentence this

has two different predicates inside of

it first of all I have if X is the POTUS

if X is the president of the United

States

that's a predicate and I will call it P

of X and then it says that if this is

going to be true then X is a US citizen

and I'll write that as a second

predicate so depending on what your X is

some X's if X has sort of lives in the

universe of people for instance if I set

my domain to be equal to just humans

some of those people are US citizens

some of those people are US presidents

and what this claim is is that all the

ones that are US presidents that set of

people throughout history all of those

people are US citizens note that when I

phrase it here as an if

as a conditional that the universal part

they for all X in the people part is

just sort of not explicitly written but

it still is they're implicitly when I

say if X is going to be the president

United States I am saying for all people

that happen to be the president of the

United States so this is sort of the

equivalence between these two ideas

saying if some predicate is going to be

true in other words if X has the

property of being the president noted

states is the same thing as saying for

all people or every person who is the

president United States they are those

people are then going to become US

citizens another way to phrase this is

sort of natural English that might make

the for all or the every more explicit

is something like this every president

of the United States is a US citizen I

haven't put it in my formal if-then

conditional form but the every that I

have here is the same thing as this for

all right every is kind of like a for

all so this is saying every president of

the United States is the same thing as

if X is the president of States is a US

citizen same thing as that X is a US

citizen

Universal Conditionals P(x) implies Q(x)

We describe the notation for universal conditionals of the form P(x) implies Q(x). This is short hand for saying that FOR EVERY specific value x, the logical implication P(x) implies  Q(x) is true. That is, this is a claim about two PREDICATES saying that the implication is true for all STATEMENTS where x is an element in the domain.

►Full DISCRETE MATH Course Playlist: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS

Other Course Playlists:

►CALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m

►CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n

►CALCULUS III: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd

►LINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6

► Want to learn math effectively? Check out my "Learning Math" Series: https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw

►Want some cool math? Check out my "Cool Math" Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho

*****************************************************
YOUR TURN! Learning math requires more than just watching math videos, so make sure you reflect, ask questions, and do lots of practice problems!

****************************************************

►Follow me on Twitter: http://twitter.com/treforbazett

BECOME A MEMBER:
►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join

MATH BOOKS & MERCH I LOVE:
► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett


in this video I want to investigate

whether I can make this particular

piecewise function continuous everywhere

well I have a little bit of freedom

notice that there's a see inside of

there so the hope is that by

manipulating the C I can make it

continuous everywhere first I want to

note this but look at this function here

or this function individually like if

this applied for all values of X either

of those are just polynomials one's a

quadratic one's a linear and polynomials

are continuous absolutely everywhere

there's no division by zero there's no

infinite spikes there's no oscillatory

behavior these are continuous everywhere

so the only possibility the only spot

where there could be a problem with this

value of x equal to one that's where it

might be that the limit from the left

and I'll limit from the right and the

function value might be different okay

so let's match them our goal is this I

want to say that the limit as X goes to

the interesting spot the 1 as X goes to

1 from let's do the left first of f of X

what I'm hoping is that this is going to

be the same thing as the limit as X goes

to 1 now from the right of my function f

of X so I need that to be the case if I

have any hope for it to be continuous

and an indeed my third condition is that

I want it to be equal to F of 1 that

one's going to come more or less for

free ok well now let's do the one from

the left now from the left is this top

one up here X values less than one so in

other words this is the same thing as

the limit as X goes to 1 from the left

of see x squared plus 1 and and this one

over here is just going to be equal to

the limit as X goes to one now from the

right of 2 X minus C and the F of 1 is

just going to be plugging in here this

is going to be equal to 2 minus C so

these are the three different things

that I need to be equal ok well how do i

do these limits both of these limits

over here are easy in the sense that

these are polynomials so that

ulis so the computer limit I just plug

it in so in this case if I plug in one

for the value of X so this is equal to

two minus C the same value as the

function value as we would have expected

and then over here I'm gonna plug in one

and I'm gonna get c plus one so this

gives me an equation it tells me that I

want C plus one to equal to minus C so

let's do that we're gonna see plus 1 to

be equal to two minus C and then if I do

a little bit of manipulation on this

okay I'm gonna move this C over and this

one I'm gonna subtract off so I'm gonna

have a two C is equal to one and so

finally I have C equal to the value of

one half

and indeed when I do this this value of

C would have hat it fixes the potential

discontinuity at one the letter from the

left is equal to limit from the right so

yes indeed this is continuous at one and

because of continuous everywhere else as

they're polynomials it's continuous

everywhere

Example: When is a Piecewise Function Continuous?

In this example, we look at a piecewise function that has a parameter. The question is, can we find a value of the parameter that makes the function continuous everywhere? Indeed, if we check that the limit from the left, the limit from the right, and the function value are all equal at the interesting point in the piece wise function, then we will get equations in the parameter we can solve.

**************************************************
Now it's your turn:
1) Summarize the big idea of this video in your own words
2) Write down anything you are unsure about to think about later
3) What questions for the future do you have? Where are we going with this content?
4) Can you come up with your own sample test problem on this material? Solve it!

Learning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master math means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK.

***************************************************
Want more ideas for learning math effectively?
►How to Watch Math Videos:
https://www.youtube.com/watch?v=Ljq5y3ksk8o

►5 Tips to Make Math Practice Problems Effective: https://www.youtube.com/watch?v=LPH2lqis3D0

****************************************************
►Want some cool math? Check out my "Cool Math" Series:
https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho

****************************************************
Course Playlists:
►Calculus I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m

►Calculus II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxdQCBjYswqbn7LxL1pW4cW4

►Discrete Math: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS

►Linear Algebra: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6

*****************************************************
►Follow me on Twitter: http://twitter.com/treforbazett
*****************************************************

This video was created by Dr. Trefor Bazett, an Assistant Professor, Educator at the University of Cincinnati.

BECOME A MEMBER:
►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join

MATH BOOKS & MERCH I LOVE:
► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett


in this video I need to show you an

effective method to be able to take one

vector and write it in terms of some

other basis to write it in some other

coordinate system and while I show you

the best way to do this kind of

computation I'm also going to introduce

a little bit of notation it's going to

help us going forward in the course so

the problem is this I have some

particular basis specified in this case

looks like the base is 1 1 and minus 2 1

then I have some vector for example the

vector 4 1 and what I want to know how

to do is how do I write that vector in

the B basis that is if I take my 4 1

what are the coefficients the C 1 and

the C 2 so that it can be written here

as a linear combination of the b one of

the B 2 now whatever you see a linear

combination I want to have a little bit

alarm bell go off in your mind because

linear combinations and multiply by a

matrix are the same thing indeed running

this linear combination is just the same

thing as saying that this is the

multiplication of the C 1 and C 2 by

these particular matrix the 1 1 minus 2

1 well what do I have I have a linear

system matrix times a vector is equal to

some other vector and I want to solve

that system and I could use some row

reductions I could use my normal trick

to solve a linear system of equations

but I want to talk about it in terms of

invertible matrices so the matrix that I

have here I can give it a name this

matrix is called P sub B this is a

matrix associated to every single basis

and you just take the vectors of the

basis and put them in as the columns of

the matrix and then the other thing to

note is that the C 1 C 2 we previously

define that to be the vector X written

in the B basis so this is just going to

be my vector X written in the B basis at

4 1 is my X and the C 1 C 2 is my X

written in the B basis now what I'm

talking about all of our two or all of

our three but the P matrix I'd get this

is going to be a square matrix we're

talking about different bases of our two

so they're all going to have two

components and there's going to be two

of them and in fact because the basis

where the vectors are linearly

independent if your row reduced is gonna

have a lead

one in every single column and so that

this matrix is going to be invertible so

I can go and reverse this process I've

got right now X is equal to a matrix

times the vector expert in the B basis I

can invert it and what I'm gonna get is

this I'm gonna get that the C 1 C 2 is

the inverse of that matrix times the 4 1

times the X and I know how to take the

inverse of a 2x2 matrix I swap the a and

the D put a minus sign in front of the B

and the C you divide out by ad minus BC

so I inverted this particular matrix and

now I can just multiply it out and what

do I get I get the value 2 minus 1

in fact we always have these two

different ways of solve systems you can

do the row reduction methodology or if

it's a square matrix you can just invert

it but the nice part about being will

invert it is now you can do it for every

single vector at once you just always

have to just do this multiplication you

don't have to repeat that row reduction

step every single time so if you can use

invertible matrices I usually suggest

that's the superior way to do it in

generality regardless I have now figured

it out my C 1 C 2 is equal to the 2

minus 1 and in doing so I actually can

come up with two sort of general

formulas the first is going to be that X

can always be written as this P matrix

times X written in the B basis or to

flip it around the other way X written

in the B basis can be the inverse of

this P matrix multiplied by n where the

where the PB matrix just has the first

and second basis vector as the first and

second call

Writing Vectors in a New Coordinate System   **Example**

The best method to algebraically express a vector in some given coordinate system (ie in terms of a given basis) is by introducing a matrix whose columns are the basis vectors, and then inverting that. We pick up some new notation along the way.

**************************************************
Now it's your turn:
1) Summarize the big idea of this video in your own words
2) Write down anything you are unsure about to think about later
3) What questions for the future do you have? Where are we going with this content?
4) Can you come up with your own sample test problem on this material? Solve it!

Learning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master math means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK.

***************************************************
Want more ideas for learning math effectively?
►How to Watch Math Videos:
https://www.youtube.com/watch?v=Ljq5y3ksk8o

►5 Tips to Make Math Practice Problems Effective: https://www.youtube.com/watch?v=LPH2lqis3D0

****************************************************
►Want some cool math? Check out my "Cool Math" Series:
https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho

****************************************************
Course Playlists:
►Calculus I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m

►Calculus II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxdQCBjYswqbn7LxL1pW4cW4

►Discrete Math: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS

►Linear Algebra: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6

*****************************************************
►Follow me on Twitter: http://twitter.com/treforbazett
*****************************************************

This video was created by Dr. Trefor Bazett, an Assistant Professor, Educator at the University of Cincinnati.

BECOME A MEMBER:
►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join

MATH BOOKS & MERCH I LOVE:
► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett


okay so let's look at one more example

here where I have a series and my series

is like really just kind of complicated

if I try to write out for example what

SN was it would just like be this

horrible mess and we don't really know

how to do anything with it

indeed this pretty much often the case

in the telescoping series we could write

out the SN and everything canceled it

was really neat but in the vast majority

of cases if we try to write out what SN

was it's just gonna be this horrifying

mess so how do we deal with this and if

I look at this I'm like very tempted to

do an integration because I see an e to

the minus N squared and then multiplied

by a power of n on the on the bottom it

kind of like screams you subs to me but

that's something we did back in

integration so let's just compare this

to the improper interval indeed if I

consider the improper integral 1 to

infinity of 4 times X e to the minus x

squared so I'm replacing my n with X

here DX

well I'd first know that this is the

same thing as the limit as T goes to

infinity of the integral from 1 up to T

of 4 X e to the minus x squared DX

that's how I deal with improper under

both I write this as the sort of this

limit of these quote-unquote partial

integrals are sort of analogous to

partial sums in series so if I do a

little bit of the u sub maybe I'm going

to say that U is equal to x squared that

means that D U is equal to 2x DX

if X is equal to 1 that implies that U

is equal to 1 and if X is equal to T

that implies that U is equal to T

squared so I can replace this entire

thing this is just the limit as T goes

to infinity of the integral from 1 up to

T squared I have a 4x so this is a 2 do

you E - the - you do u and this is just

the limit as T goes to infinity of a

minus sign because the e to the minus u

T e to the minus u evaluated between 1

and T squared which is this

same thing as limit as T goes to

infinity of minus 2 e to the minus T

squared all subtracted because I've got

two minus signs I go put a plus sign all

plus 2 e to the minus 1 and then if I

evaluate this as T goes to infinity this

this whole first term is going to zero

so just left with this 2 over e and

therefore that's my answer so that's my

answer for the integral the integral

converges to 2 over e and therefore my

original series from 1 to infinity of 4

and E the minus N squared converges and

again note that I'm not saying it

converges to 2 over e it almost

certainly doesn't I'm merely saying that

it converges

Integral Test - Bonus Example

Learning Objectives:
1) Understand the concept of a remainder
2) Follow the geometric derivation of estimating the remainder by integrals
3) State the Remainder Estimate
4) Apply the estimate to determine how many terms are needed to achieve a specific level of uncertainty.

This video is part of a Calculus II course taught at the University of Cincinnati.


कर दो

220 प्रेजेंटेशन ऑन वर्ड्स स्टार्टिंग विद

न्यू चैप्टर इन दैट इज रिलेटेड क्वांटिटी

अफेयर्स एंड स्पोर्ट्स क्लब प्रेजेंटेशन

आई विल इंट्रोड्यूस यू टो थे कॉन्टेक्स्ट

आफ नेक्स्ट पॉइंट पांच वर्षीय स्टार्टिंग

विद डेफिनेशन आफ व्हाट इज बेस्ट क्वालिटी

50 ईयर्स और से ट्यूब इंट्रस्टिड इफ वन इज

दिस को पॉजिटिव यदि को 1968 बिन अधिकतम इन

विन 10 को ऑफ द कंट्री फॉर एग्जांपल ए

गुरु कैन सी हाउ टू मेक एग सब्सक्राइब

सीधे subscribe The Channel Please

subscribe and

ए स्कोप ऑफ दिस कंट्री फॉर उस वे कैन

रिप्लेस थिस प्रपोजिशन फंक्शन त्यौहार

प्रपोजिशन फंक्शंस पे थिक प्रॉब्लम सॉल्व

इन दिशाओं स्पॉइल्स पे राइटिंग ऑल

सबस्क्राइब कोंबिनेशन सब्सक्राइब

कंसीडरिंग ऑल सबस्क्राइब अकॉर्डिंग्ली थे

डिफरेंट कांबिनेशंस फॉर ऑल लाइक यू ऑल

सब्सक्राइब टो

प्लेन मोड ऑफ करंट अफेयर्स फॉर द फॉरेस्ट

द्वेलर्स लॉस्ट इन थे वेस्ट ऑयल video.com

30 मिन क्राफ्ट फॉर वाइफ इन प्लेस फॉर ऑल

सब्सक्राइब बटन क्लिक करें और मेरे चैनल

को सबस्क्राइब करना न भूलें और

यह सब कुछ लुट लो

कर दो

[संगीत]

Introduction to Nested Quantifiers

Discrete Mathematics: The Introduction to Nested Quantifiers
Topics discussed:
1) The definition of nested quantifiers.
2) The example of nested quantifiers.
3) Different combinations of nested quantifiers.

Follow Neso Academy on Instagram: @nesoacademy(https://bit.ly/2XP63OE)
Follow me on Instagram: @jaspreetedu(https://bit.ly/2YX26E5)

Contribute: http://www.nesoacademy.org/donate

Memberships: https://bit.ly/2U7YSPI

Books: http://www.nesoacademy.org/recommended-books

Website ► http://www.nesoacademy.org/
Forum ► http://forum.nesoacademy.org/
Facebook ► https://goo.gl/Nt0PmB
Twitter      ► https://twitter.com/nesoacademy

Music:
Axol x Alex Skrindo - You [NCS Release]

#DiscreteMathematicsByNeso #DiscreteMaths #NestedQuantifiers


the even and all of integers are

familiar concepts to us however in this

particular video what we're going to do

is we're going to try to define exactly

what we mean by the even and odd

integers we're going to try to have the

most precise and formal and rigorous

definition of these concepts that we can

have and the idea is this when we're

trying to make claims about mathematics

and when we're trying to prove claims in

mathematics we need to have a high

degree of precision and simply saying we

understand what even numbers are their

numbers like two four six eight and

seeing that we understand what odd

numbers like three five seven and nine

are is insufficient we need to use

incredibly precise ideas for what these

concepts actually are now when I look at

this in formal definition there's a sort

of existential nature to it when we say

that an integer say n can be written as

twice something really what we're saying

is that there exists this existential

claim there exists some other integer K

where the N can be written as twice at

so if I want to take this informal

definition and make it into a formal one

I might write something like this I

would say that n is an integer if there

exists another integer K so so there

exists a K any integers but has the

property that the N is twice K now why

do I like this formal definition so much

it's because now there's actually an

equation here N equals to K and I know

something about what K is K is an

integer

so if I want to make a proof that

involves the even integers now I can

actually substitute it in I can write

equation I have a precise way of writing

what I mean when I talk about an even

integer ok same story but for the

auditors well okay what's an odd integer

informally odd integers are the integers

that are just not even as in if two four

six eight are even integers then the

other ones though the one three five

seven

those are the auditors

you are allowed to define something as

being not some other thing okay but

let's try to be a little bit more

precise about this if I take say six we

said that was even because it was twice

an integer but what about seven so seven

is odd but what I notice about seven is

that it's twice an integer twice three

plus one so all of the odd integers can

really be written as twice an integer

plus one so I maybe use that as my

formal definition again it's an

existential claim I'm saying there

exists a cave in the integers such that

your audience a Duran can be written as

twice K plus one and now again I have a

precise formal definition of what I mean

by all integers there's an equation here

I can send it to that equation into

other equations that I can use these

definitions to make proofs about the

even and odd integers and indeed in the

next video that's exactly what I'm going

to do I'm going to do our first proof

and I'm going to use these definitions

of even and odd in the proof that I'm

going to do in the next video

Formal Definitions in Math | Ex: Even & Odd Integers

We've all seen even and odd integers before. But how - exactly - are they defined? How would you use them in a proof about the even, and odd integers. In this example we properly define the even and odd integers. In the next video, we will use them in a proof about even and odd integers.

►Full DISCRETE MATH Course Playlist: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS

Other Course Playlists:

►CALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m

►CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n

►CALCULUS III: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd

►LINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6

► Want to learn math effectively? Check out my "Learning Math" Series: https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw

►Want some cool math? Check out my "Cool Math" Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho

*****************************************************
YOUR TURN! Learning math requires more than just watching math videos, so make sure you reflect, ask questions, and do lots of practice problems!

****************************************************

►Follow me on Twitter: http://twitter.com/treforbazett

BECOME A MEMBER:
►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join

MATH BOOKS & MERCH I LOVE:
► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett


before you learn about Rings you should

really learn about groups so if you

haven't already watched our group video

then hit the pause button and check it

out all wait go

on roughly speaking a ring is a set of

elements where you can add subtract and

multiply but you may not be able to

divide and multiplication may not be

commutative for example the 2 x two

matrices with real numbers form a ring

and matrix multiplication is not

commutative by not requiring a

multiplication to be commutative Rings

generalize things like matrices which

are pretty important by the way did you

notice I said elements instead of

numbers H that's because some rings

consist of numbers While others are made

of more exotic objects like pols or

matrices before we give a more precise

definition let's see some

examples the integers are a ring you can

add subtract and multiply LLY any two

integers and you'll get another integer

but can you divide two integers H well

yes but you may not get an integer if

you divide 3 by five you get a fraction

not an integer that's key if you add

subtract or multiply two elements in the

ring you get another element in the ring

we say the ring is closed under addition

and multiplication what happens in the

ring stays in the

ring another example of a ring is the

set of polinomial with integer

coefficients you can freely add subtract

and multiply two such polinomial and you

will get a third polinomial of this type

but division no I don't think so if you

divide x +1 by x -1 you get x +1 over x

-1 hey it's a thing but it's not a

polinomial so now for the official

definition a ring is a set of elements

with two operations addition and

multiplication if you add two elements

in a ring you get another element in the

ring but earlier I said you can add and

subtract in a ring what gives well

subtraction is just the same as adding

inverses so when I said you can add and

subtract I was essentially saying that

rings have negative elements and if you

add an element to its negative you get

zero so in a ring the elements are a

group under addition better yet addition

is commutative in

rings now here's where things get

serious if you multiply two elements in

a ring you get another element in the

ring it's close under multiplication no

surprise there but elements may not have

inverses under multiplication we may not

be able to divide so while the ring is a

nice aelan group under addition things

may not be so nice with multiplication

but hey at least multiplication is

associative and to wrap things up

there's a rule which connects addition

and multiplication the distributive rule

all the numbers you studied in

arithmetic obey the distributive rule

the expressions in algebra do as well so

we're good

this isn't something new ladies and

gentlemen I give you the

rank

Abstract Algebra: The definition of a Ring

Learn the definition of a ring, one of the central objects in abstract algebra.  We give several examples to illustrate this concept including matrices and polynomials.

Be sure to subscribe so you don't miss new lessons from Socratica:
http://bit.ly/1ixuu9W

♦♦♦♦♦♦♦♦♦♦

We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition
http://amzn.to/2oOBd5S

Milne, Algebra Course Notes (available free online)
http://www.jmilne.org/math/CourseNotes/index.html

♦♦♦♦♦♦♦♦♦♦

Ways to support our channel:

►  Join our Patreon : https://www.patreon.com/socratica
►  Make a one-time PayPal donation: https://www.paypal.me/socratica
►  We also accept Bitcoin @ 1EttYyGwJmpy9bLY2UcmEqMJuBfaZ1HdG9

Thank you!

♦♦♦♦♦♦♦♦♦♦

Connect with us!

Facebook: https://www.facebook.com/SocraticaStudios/
Instagram: https://www.instagram.com/SocraticaStudios/
Twitter: https://twitter.com/Socratica

♦♦♦♦♦♦♦♦♦♦

Teaching Assistant:  Liliana de Castro
Written & Directed by Michael Harrison
Produced by Kimberly Hatch Harrison

♦♦♦♦♦♦♦♦♦♦


one kind of funny statement is a toogy a

toogy and we we usually use T for that

opposed to P or Q or R we use the symbol

t for a topology this is the statement

that is always true and can never be

false so for instance consider the

statement that dog is a mammal well that

statement is always true doesn't matter

which dog you're referring to in the

concept of what a dog is implicitly is

the concept that it is going to be a

mammal that is one of the properties of

being a dog so if I if I have the

statement that dog is a mammal it does

not matter which dog I'm referring to so

this statement is always true

irregardless of my inputs so this is

sort of play some funny games when it

comes to looking at what the truth

tables are so I'm going to imagine that

I've got the statement I want to look at

what T or p is where I've got a toy T

and A ology it's truth table is very

simple it's just always true right the

the topology can never be false so it's

just true true true true everywhere and

then I also have this other statement p

p is just a normal old statement here so

it can be true or false so maybe its

first possibility will be true and a

second possibility will be

false and then if I want to take a

topology or some statement remember or

is true if if one of the two are going

to be true but the ology is always true

so it's always the case that one of them

is going to be true here so there's two

truths there which is good enough to

give me a true here there's only one

true here but that's fine it's an or

statement I only need one of them to be

true so that's going to be true there as

well I can also have contradictions

contradictions are labeled by a c now

for contradiction and this is a

statement that is always false so this

is an example that dog is a reptile

again in the notion of a dog it is

implicit that dogs are mammals and not

reptiles so it can never be true that

that dog is a reptile so this means that

in our truth table whenever I have a c

it is going to be a column of all

falsehoods so here I'm going to have a

false and here I'm going to have a false

and then I've also done a simple example

I'm trying to look at C and P where p is

just some other statements so let's fill

in a couple different possibilities for

p could be true and it could be false

and then if I want to look at my final

conclusion which was C and P remember

and requires both of them to be true but

because we have my contradiction always

been false it will never be the case

that both of them are true and so since

there's one false there there's going to

be a false there since there's a false

there there's going to be false there as

well so when I had an or statement with

a a a topology it forced that to be toy

it made it always going to be true if

you if you took or a toy you could just

take the value of the toy would be

always true if you take and a

contradiction then it's just going to

make it into its own contradiction which

is going to be all falses in other words

I can say that that c and P is a

contradiction it will always be false as

this table I mentioned and as we saw in

the previous slide if I take t or P that

is always true and so it is always a

topology

Tautologies and Contradictions

Learning Objectives:
1) Tautologies: examples and truth table
2) Contradictions: examples and truth table
3) Use tautologies and contradictions in compound statements.

****************************************************
►Full Course Playlist: DISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS
****************************************************
Other Course Playlists:
►CALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m

►CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n

►LINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6
***************************************************
► Want to learn math effectively? Check out my "Learning Math" Series: https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw

►Want some cool math? Check out my "Cool Math" Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho

*****************************************************
YOUR TURN! Learning math requires more than just watching videos, so make sure you reflect, ask questions, and do lots of practice problems!
****************************************************
►Follow me on Twitter: http://twitter.com/treforbazett
*****************************************************

This video was created by Dr. Trefor Bazett