Mathematical Proof Writing

Summary

El video ofrece consejos para aprender a escribir demostraciones matemáticas, destacando la importancia de comprender la estructura de las pruebas para disfrutar de las matemáticas. Se aborda la dificultad inicial y cómo, una vez dominada, transforma la experiencia de aprendizaje. Se recomienda tomar clases, especialmente con buenos profesores que escriben demostraciones claras, y se sugiere la exploración de múltiples recursos, incluyendo libros, cursos en línea y videos de YouTube. El video enfatiza la necesidad de feedback para mejorar y menciona varios libros que pueden ser útiles en el proceso de aprendizaje.

Takeaways

• 😀 Aprender a escribir demostraciones matemáticas es crucial para comprender y amar la matemática a un nivel más profundo.
• 🧠 Es normal sentirse desanimado al aprender demostraciones, pero la clave es entender la estructura y la lógica detrás de ellas.
• 📚 Tomar clases, especialmente con buenos profesores que escriben demostraciones claras, es una de las mejores formas de aprender a escribir pruebas.
• 💻 Los cursos en línea y los videos de YouTube pueden ser útiles para aprender demostraciones, pero la retroalimentación es esencial para el progreso.
• 📖 Leer y trabajar con múltiples libros de demostraciones puede ayudar a mejorar la habilidad de escribir pruebas, ya que cada libro puede ofrecer diferentes perspectivas.
• 🤓 Es importante aprender no solo a copiar demostraciones, sino a comprender cómo construir las propias para desarrollar un pensamiento lógico sólido.
• 👨‍🏫 Los profesores suelen elegir libros de matemáticas que no tienen respuestas para facilitar la asignación de problemas como tareas y exámenes.
• 🎓 Dominar la escritura de demostraciones es fundamental para obtener un título en matemáticas y avanzar en áreas más complejas del aprendizaje matemático.
• 📝 La escritura de demostraciones mejora con el tiempo y la práctica, y es un viaje de aprendizaje continuo en el campo de las matemáticas.
• 💡 Incluso después de aprender a escribir demostraciones, los desafíos matemáticos siguen siendo difíciles, pero se vuelve más manejable con una sólida base en la lógica y la demostración.

Q & A

¿Qué hace que las pruebas matemáticas sean un punto de inflexión en el aprendizaje de matemáticas?

Las pruebas matemáticas son un punto de inflexión porque permiten a los estudiantes ver la lógica detrás de los problemas y construir un razonamiento sólido. Al dominar las pruebas, se puede comenzar a apreciar la belleza y la estructura de la matemática, lo que lleva a un mayor amor por la disciplina.

¿Por qué a menudo se siente que las pruebas matemáticas son difíciles de aprender?

Las pruebas matemáticas pueden ser difíciles de aprender porque requieren una comprensión profunda de la lógica y la estructura matemática. Muchas personas no llegan a este nivel debido a la falta de entendimiento o la falta de una buena guía, lo que puede hacer que el proceso parezca abrumador y desalentador.

¿Qué consejo se da en el video para aquellos que están luchando con el aprendizaje de pruebas matemáticas?

El video sugiere que es normal sentirse desanimado al aprender pruebas matemáticas y que el entendimiento es clave. Se recomienda ser persistente y buscar la mejor guía posible, ya sea en forma de clases, libros o recursos en línea.

¿Por qué es importante aprender a escribir pruebas en matemáticas?

Aprender a escribir pruebas es fundamental en matemáticas porque proporciona la base para entender y explorar conceptos más avanzados. Permite a los estudiantes construir un conocimiento sólido que les permite abordar temas complejos y avanzar en su educación matemática.

¿Qué recursos se recomiendan en el video para aprender a escribir pruebas matemáticas?

El video recomienda utilizar libros, clases en línea, cursos universitarios y videos de YouTube como recursos para aprender a escribir pruebas. Se sugiere obtener la mayor cantidad de libros posibles para ver diferentes enfoques y se mencionan específicamente varios títulos de libros de pruebas.

¿Cuál es la importancia de la retroalimentación al aprender a escribir pruebas matemáticas?

La retroalimentación es crucial al aprender a escribir pruebas porque permite a los estudiantes entender sus errores y mejorar su enfoque. Es difícil obtener retroalimentación de libros, por lo que se recomienda tomar clases donde los profesores puedan proporcionar esta retroalimentación valiosa.

¿Qué libro gratuito se menciona en el video para aprender a escribir pruebas?

El libro gratuito mencionado en el video es 'Book of Proof' de Hammock, que se puede encontrar en línea y es una buena introducción al tema de las pruebas matemáticas.

¿Cómo se puede mejorar la habilidad para escribir pruebas matemáticas según el video?

El video sugiere que la mejor manera de mejorar la habilidad para escribir pruebas es a través de la práctica constante, el aprendizaje de profesores experimentados y la observación y emulación de pruebas limpias y claras.

¿Qué es la estructura de una prueba matemática y por qué es importante?

La estructura de una prueba matemática es una serie de pasos lógicos y deductivos que llevan de un enunciado a una conclusión. Es importante porque permite a los matemáticos demostrar teoremas y conceptos de manera rigurosa y comprensible.

¿Cuál es la relación entre el aprendizaje de pruebas y la obtención de un grado en matemáticas?

El aprendizaje de pruebas es esencial para obtener un grado en matemáticas porque la mayoría de las clases avanzadas en matemáticas requieren la capacidad de entender y construir pruebas. Dominar esta habilidad abre las puertas a áreas más especializadas y avanzadas del estudio matemático.

Outlines

00:00

📚 Aprendiendo a escribir demostraciones matemáticas

El primer párrafo enfatiza la importancia de comprender y amar las demostraciones matemáticas al avanzar en el aprendizaje de matemáticas. Se menciona que la mayoría de las personas no llegan a este nivel por varias razones, pero el vídeo pretende ofrecer consejos para lograrlo. Se aborda la dificultad inicial de aprender a escribir demostraciones y cómo, una vez que se comprende la estructura, se puede disfrutar y apreciar la belleza de la lógica matemática. Se sugiere que el aprendizaje de la escritura de demostraciones es crucial para dominar temas avanzados de matemáticas y que, a pesar de ser difícil, es fundamental para el crecimiento en el campo.

05:01

👨‍🏫 La importancia de un buen profesor en el aprendizaje de demostraciones

Este párrafo relata la experiencia personal del narrador con dos profesores excepcionales que le enseñaron a escribir demostraciones de manera diferente, lo que resultó en un gran beneficio para su comprensión. Se discute la ventaja de tomar clases con profesores que escriben demostraciones claras y cómo el aprendizaje de la escritura de demostraciones se ve afectado por la calidad de la enseñanza recibida. Además, se menciona la dificultad de obtener retroalimentación de libros en comparación con la experiencia de aprender en un entorno educativo donde se recibe retroalimentación directa.

10:02

📚 Mejores recursos para aprender a escribir demostraciones

El tercer párrafo aborda la selección de libros y recursos para aprender a escribir demostraciones, destacando la importancia de aprender de las mejores fuentes posibles. Se mencionan varios libros recomendados por la comunidad y por ex profesores, como 'Book of Proof', 'How to Read and Do Proofs' de Daniel Solow, 'Proofs' de Jake Cummings, y otros. El narrador comparte su experiencia con estos libros y cómo cada uno aporta algo diferente al aprendizaje del proceso de demostración. Se enfatiza la necesidad de tener múltiples recursos para abordar diferentes enfoques y estilos de demostración.

15:03

🎓 La conexión entre la escritura de demostraciones y la obtención de un grado en matemáticas

El último párrafo enfatiza la conexión entre el dominio de la escritura de demostraciones y la capacidad de completar un grado en matemáticas. Se menciona la importancia de la retroalimentación en el proceso de aprendizaje y cómo, a pesar de los desafíos, el éxito en la escritura de demostraciones abre puertas para estudiar temas avanzados. El narrador comparte su propia experiencia con la escritura de demostraciones y cómo la toma de múltiples cursos en demostraciones lo ayudó a superar los retos iniciales y avanzar en su educación matemática.

Keywords

💡Pruebas matemáticas

Las pruebas matemáticas son demostraciones lógicas utilizadas para confirmar la veracidad de una afirmación matemática. En el video, se enfatiza la importancia de aprender a escribir pruebas como un paso fundamental para apreciar la belleza y la lógica de las matemáticas. El orador comparte su experiencia de cómo el aprendizaje de las pruebas transformó su percepción sobre las matemáticas, haciéndolas parecer 'hermosas' y 'asombrosas'.

💡Estructura de las pruebas

La estructura de las pruebas es el marco que guía el proceso de demostración en matemáticas. El video menciona la necesidad de comprender la estructura para escribir pruebas efectivas, como las pruebas directas, pruebas por contradicción y pruebas por contraposición. Esta comprensión es crucial para abordar temas avanzados de matemáticas.

💡Convergence test

El test de convergencia es un método utilizado para determinar si una serie infinita converge o diverge. Aunque no es el foco principal del video, se menciona como ejemplo de un concepto matemático que el orador disfrutó antes de adentrarse en el mundo de las pruebas, destacando la evolución de su pasión por las matemáticas.

💡Lógica

La lógica es el estudio de la validez de las formas de razonamiento, y es fundamental en la escritura de pruebas matemáticas. El video sugiere que el entendimiento de la lógica es esencial para construir pruebas sólidas y es una habilidad que se perfecciona con el tiempo y la práctica.

💡Clases de matemáticas

El video recomienda tomar clases de matemáticas, especialmente aquellas que se centran en la escritura de pruebas, como una de las mejores formas de aprender. Las clases proporcionan la oportunidad de ver a profesores experimentados escribiendo pruebas y ofrecen retroalimentación valiosa para el aprendizaje.

💡Feedback

El retroalimentación es un elemento crítico mencionado en el video para el aprendizaje de la escritura de pruebas. Es la forma en que los estudiantes pueden aprender de sus errores y mejorar su habilidad para escribir pruebas. El orador sugiere que el acceso a un profesor que pueda proporcionar esta retroalimentación es esencial.

💡Libros de matemáticas

Los libros de matemáticas son una herramienta recomendada en el video para aprender a escribir pruebas. Se mencionan varios títulos específicos, como 'How to Read and Do Proofs' y 'Book of Proof', que ofrecen ejemplos y ejercicios para practicar la escritura de pruebas. El orador también destaca la importancia de tener múltiples recursos para abordar diferentes enfoques y estilos de explicación.

💡Curso en línea

Los cursos en línea son presentados en el video como una alternativa viable para aprender la escritura de pruebas, especialmente para aquellos que no pueden asistir a clases presenciales. Estos cursos ofrecen la ventaja de la flexibilidad y, en algunos casos, la acreditación académica.

💡Estrategias de aprendizaje

El video discute varias estrategias de aprendizaje para la escritura de pruebas, incluyendo la imitación de pruebas de profesores experimentados, la práctica con múltiples recursos y la búsqueda activa de retroalimentación. Estas estrategias son presentadas como formas efectivas de superar los desafíos inherentes al aprendizaje de la escritura de pruebas.

💡Matemáticas avanzadas

Las matemáticas avanzadas son un tema que se discute en el video como un área donde la habilidad para escribir pruebas es esencial. Se mencionan campos específicos como el álgebra abstracta y el análisis real, donde la capacidad de escribir pruebas es crucial para comprender y contribuir al conocimiento matemático.

Highlights

Understanding how to write mathematical proofs is crucial for appreciating the beauty of mathematics.

Many people struggle with proofs, but it's a normal part of learning and it gets better with understanding.

Learning to write proofs can be a game-changer, transforming your view of mathematics.

The logical flow of ideas in proofs is often described as beautiful by those who understand it.

Proof writing can feel defeating when you don't understand the structure, but it's okay to feel that way.

Learning proof writing is essential for advancing in mathematics.

Different types of proofs such as direct, by contradiction, and by contrapositive are important to learn.

Having a proof-based background allows you to attempt more challenging subjects in mathematics.

Proof writing skills evolve over time, and it's still difficult even after mastering the basics.

Learning from the best sources is crucial when it comes to writing proofs.

Books are a good way to learn proof writing, but they may not provide the feedback you need.

Taking a class, especially with a good professor, is one of the best ways to learn to write proofs.

Online courses can be a good alternative for learning proof writing with the benefit of feedback.

YouTube videos and websites like mathsorcerer.com offer resources for learning proof writing.

Books like 'Book of Proof', 'How to Read and Do Proofs', and 'Proofs' by Jake Cummings are recommended for learning.

It's beneficial to have multiple proof books to compare and contrast different explanations and proofs.

Once you learn to write proofs, you can tackle higher-level math classes and potentially earn a math degree.

The biggest hurdle in learning proof writing is getting feedback on your proofs.

The speaker shares personal experiences with learning proof writing and the impact of having two courses simultaneously.

Transcripts

once you learn how to write mathematical

proofs once you actually understand how

to do it so you can look at a problem

and you can say okay this is what I

assume this is what I have to show and

then in your mind you have a couple

ideas once you get to that level that's

when it really takes off that's when you

really start loving mathematics

most people don't get to that level for

a variety of reasons and so in this

video I'm going to give you some tips

for getting to that level

also if you are trying to get to that

level right now

and you're struggling to learn the right

proofs and you hate it

just know that it's okay

it's normal

and I really think that the reason that

maybe you don't like it is simply

because you don't understand it right so

because you don't get it but once you

get it okay once you get it it's a game

changer and like just it's beautiful

you know I I thought I liked math when I

was in calc 2. I loved infinite series

it was so cool I love the convergence

test I was like oh this is so cool the

p-test the ratio test I was such a such

a nerd you know I loved it but when I

learned to write proofs

that was a game changer I was like okay

this is beautiful this is this is

amazing this logical flow of ideas is

just beautiful

but when you don't understand the

structure

when you don't know how to do it it can

make you feel defeated

and it's it's okay to feel that way

but if you really want to take your math

to the next level you have to learn to

write proofs and trust me once you

understand the structure

it's not that bad it's it's learning new

math that's still hard but having that

structure behind you having the proof

structure knowing how to construct

proofs knowing how to write a direct

proof a proof by contradiction a proof

by contrapositive knowing how to prove

and if and only if statement once you

know how to do those things you can pick

up a book on advanced mathematics and

you can start learning you can pick up a

book on an abstract algebra and start

learning abstract algebra you can pick

up a book on real analysis and start

learning real analysis it's still going

to be really really hard right but you

have the proof based background you have

that Foundation that at least lets you

attempt these more challenging subjects

and and your proof writing will evolve

over time you'll get better over time

it's not just like oh you learned to

write proofs and then all of a sudden

it's super super easy no no no it's

still incredibly difficult math is a

lifelong journey of learning

so in this video I want to talk about

the best way to learn to write proofs

so first let me say something that I

didn't want to say but I'm going to say

it when it comes to writing proofs you

really want to make sure that you learn

from the best possible sources

so I would recommend that you get some

books and we'll talk about some books

later books are pretty good

but once you know how to write proofs

even the proofs and books sometimes

don't seem that good because you feel

like your own proofs are better when you

feel like your proofs are more clean and

more clear than the ones in the books

that's how you know you've reached a

certain level of like yeah I can write

proofs I know what's going on it's kind

of like being a programmer if you are

reading someone else's code you're like

okay oh that's what they did there oh

okay I understand but when you write

your own code it's like oh yeah this is

my program I like the way I wrote it

this is this is awesome this is the best

program because it makes sense to me and

I wrote down you know this code that

creates this program and this is how it

all fits together and I know it fits

together in a logical way because I

created that logical construction the

same thing happens with proof writing

you look at the proof and you say this

is my proof I constructed this proof

there was no better proof than my proof

the proof in this book is inferior that

that it's kind of a weird feeling but

when you create something it makes more

sense to you and that that's the level

you want to get to

so you want to learn from the best

if possible so a book is is a good way

now the best way to learn to write

proofs is taking a class and a class in

college face-to-face class if you can

and hopefully you can get a good

professor hopefully your professor is

good and writes clean proofs it's not

always the case it's just the way life

works but having a good professor

that can write good proofs is an

absolute requirement almost for most

people to learn to write proof and you

can learn on your own it's tough you can

do it if you get a lot of books you can

learn but

having a professor is good I was really

lucky I was very very lucky because I

was doing math and computer science so I

took a proof writing class and the

discrete math class which is a computer

science math class for computer science

students and in the discrete math class

we had to do proofs

and it was kind of cool because in my

math class we did logic and proof we

spent a lot of time on the logic and we

didn't really get to the formal proofs

or the rather the informal proofs until

the end of the semester

so we spent a lot of time on logic and

then in my discrete math class that

class covered a lot of topics so when we

got to proofs it was like okay here's

the proof here's how you do it you know

so so it was like this weird mix so I

felt like I was on top of the world

because I was getting instruction from

two very very good professors one uh

being a man from the U.S he passed away

several years ago uh great man he used

to anoint us before class very good that

was the math professor and then uh for

my computer science based math class uh

it was this Middle Eastern man he had a

very thick accent and he was really

hardcore amazing Professor so having two

distinct professors teach me proofs in

different ways was a big benefit for me

and then I had other great professors in

abstract algebra and stuff that really

really showed me how to write elegant

proofs and I tried to mimic what they

did so taking a class

lets you see experienced professionals

write quality proofs and the more math

classes you take the more proofs you see

and so what you want to do is you want

to try to emulate what you think is the

cleanest proof from these math

professors from these various classes

and you want to try to absorb that and

incorporate it into your own proof

writing

and that's how you learn I'm getting

Goosebumps I mean that that's really how

you learn to write proofs that's the

best way

now I realize that's not the easiest way

many of you you know don't want to go to

college many of you don't have the time

or the money or the energy and I get it

right time is the most valuable resource

we all have so so what can you do

besides that well you can take an online

course there's online courses at

colleges that's also good and at least

with an online course at a college you

get college credit there's

accountability but more importantly you

get feedback

feedback is something that is very hard

to get from a book okay as much as I

love books and I think you should buy

books and books are great it's hard to

get feedback on your own proofs from a

book

and it's hard to email people online and

get feedback you know like if someone

emails me and they send me their proof

it's hard for me to it's hard for me to

read all the emails I get but you should

email me if you have questions but I

still struggle to keep up right I don't

get to read all of them so if someone

were to send me a math proof and say can

you check my proof I probably wouldn't

have time to look at it right it takes a

lot of work time and energy

so by taking a class you have a

dedicated Professor whose job is to

actually give you that time and energy

to look at your proof so at least when

you turn in your homework assignments at

least they're graded and so when you get

them back you have some feedback on your

work and that feedback is critical it's

critical for learning to write proofs so

again that that's the best way to learn

to write proofs take a class so you get

that feedback

a lot of people they'll take a class on

proof writing and they'll get that

feedback and they'll give up

and I get it I get it I understand it's

hard right it's not easy but just know

that once you get over that hurdle

once you learn to write proofs all of

math opens up for you

so let's say you don't want to take

courses online or out of college what

can you do

well there's YouTube videos I have a

bunch of YouTube videos on proof writing

and I'm pretty sure all of my proofs are

100 correct there's no mistakes so you

can check those out I have a bunch of

playlists I also have courses on my

website mathsorcerer.com check it out I

have a couple courses on proof writing I

don't have a dedicated course for proof

writing yet but I have like two courses

that are okay and they have some decent

proofs and they're organized

let's talk about books because I think

books are a good way

let's start with the free book A book

that's going to cost you nothing it's

free you can just go on Google and

search for it

it's this one here you might say if it's

free why do you have it well because I

bought it

[Laughter]

I I read I was reading the book online

I'm I'm okay with ebooks I prefer

physical books so I went on Amazon and I

bought this I bought it on Amazon by the

way I'll leave links in the description

to all of these books this one's called

book of proof

I've read maybe a few chapters from this

book and portions of other chapters and

I've done maybe 30 40 problems from this

book

it's pretty good it's pretty good it's

it's one of the better books it's free

it's by hammock it's called book of

proof I'll leave a link in the

description

another really really good one well I'll

save this one for last I'm Gonna Save

The Best For Last just to make it

suspenseful so another really good one

which is recommended to me by one of my

former College professors is how to read

and do proofs by Daniel solo I know a

few of you here on the channel have

talked about this book you've used this

book you're working through the book and

you like it so

other people have spent some time with

this book I've read small portions of

this book I haven't spent a lot of time

with it

um but it's highly acclaimed and highly

praised by people here on the channel

and by the reviews on Amazon really good

book

another book which I've spent some time

with which is a relatively new book

which again a lot of people here on the

channel have commented on is

um proofs by Jake Cummings I know some

of you here have been working through

this book diligently I know I've seen a

lot of the comments people saying

they've been working through this book

every day and just working really hard

and they really love this book so it's a

thick book it's a little bit different

it's a little bit wordy

um

I mean I think it's pretty good it's a

very thick book so and it's inexpensive

so it's inexpensive it's worth trying my

advice is to get every single proof book

you can afford okay because proof

writing is hard

this one is one that I've had for a long

time and I've spent a long time with

this is a transition to advanced

mathematics by Chartrand palominium

Zhang this was recommended to me by a

dear friend of mine who passed away

actually

um seven months ago it's kind of sad but

he was a big fan of this book and yeah

great book I love this book I've done a

lot of the examples from this book I've

worked through them I've done a lot of

the exercises this is a quality textbook

it's a little bit on the pricey side but

it's very very good I think it's better

than

um all the ones I showed you I do think

it's better I think it's a better book

so very good book

and then we have this one here I

actually use this one to teach an

independent study with some students a

few times in college

and it's pretty good some of the

students I had

um they didn't like this book as much as

the shartran book that I just showed you

they thought this one was inferior I

think this one's great that's just my

opinion it's got some nice proofs it's

got a lot of topics right so it's a

proof book but it has all kinds of

topics in it all kinds of math in it so

definitely recommend this one if you can

afford it it's a bit on the pricey side

the last one I'll show you then we'll

talk a little bit more about proof

writing is this one here it's called how

to prove it a structured approach this

is probably currently my favorite one

because I don't know I like the size and

it was forced upon me some people were I

don't remember who it was but

um they kept leaving comments did you

get the book did you get the book I'm

like fine I'll buy it and I didn't want

to buy it because I didn't want to spend

the money I know I paid over 20 for it

and I have a lot of books and I'm not

rich so I try not to try to buy too many

books plus I have books on the floor you

can't see them I don't have any more

bookshelves so I need to get another

bookshelf

so I have I have a problem I have too

many books

great book The reason I like this book

is because he spends a lot of time on

the logic so he'll do the informal proof

Daniel Bellman who's the author and then

he'll he'll talk about the proof

multiple ways we'll try to explain it to

you

like and he'll try hard so I think

that's important when you're learning to

write proofs because you might not

always understand the explanation

and that's the key point with all of

these books right you you get one of

these books and

um you read the explanation and you

don't get it so what do you do you read

another book that's why I think it's

good to get as many books as you can and

it's one of the reasons I have so many

books right the reason I have so many

books is because math is hard people

always think that oh you know you spent

a lot of money on books I did I did but

it's been over almost 20 years I've been

collecting math books

for almost 20 years I think it might

actually be exactly 20 years I don't

know I don't want to date myself but

it's almost 20. it's almost 20 years of

just doing mathematics so yeah

pretty cool stuff

so proof writing it's a tough one

um taking a course is the best way it

really is but

it's not a guarantee that you're going

to learn right because again I'm not one

to blame other people but like you

really want to have someone who's

writing really clean proofs so that you

can mimic those proofs

I I want to pump myself here but this is

free so if you go on my channel and I

have a playlist on set theory look at my

set theory proofs those proofs are clean

they're correct they're very detailed

there's no ambiguity there's no mistakes

right I'm wearing a a stupid wizard hat

they're kind of old videos the quality

is not great I think I made these videos

before I even knew how to edit I mean I

was using a very cheap phone and I'm

still recording this with a very cheap

phone this is a very old iPhone

but the point is the mathematics is

correct okay it's correct and the proofs

are clean and you need that you need

that a lot of books have proofs and

they're very tersed they're very quick

for example um there's a famous book on

mathematical analysis it's called

principles of mathematical analysis by

Walter Rudin this is a very famous book

I I own the first edition because I'm a

collector of math books and I think it's

a great book in fact I might even sit

down and and read a little bit today you

know it's a fantastic book but it's

extremely hard to read and he doesn't

explain the proofs I mean he just says

here's Epsilon here's Delta here's the

proof how do I figure out what you

figured out it's up to you and most of

the time it's like the proof is left as

an exercise to the reader

I don't know why they do that I think

they just want you to struggle and learn

because I think that's the best way to

learn to write proofs it's true in order

to learn to write proofs you can't just

copy someone's proof you actually have

to understand how to construct your own

proof so you actually have to put in the

effort you actually have to put in the

work and I think that a lot of these

math people these mathematicians who

write these books they know that

and that's why they purposely leave out

answers also a lot of times answers are

left out because professors will take

those exercises and assign them as

homework questions and sometimes what

they'll do is they'll assign them as

test questions which is really good I

had a partial differential equations

teacher who we used a really hard book

was I thought it was harder one by

Strauss and he would give his homework

and I remember going to his office and

he wouldn't help me with the homework it

was really weird he was like oh no I am

busy oh yeah this is just waving his

hands and he had some weird smell coming

from his office it was just a really

weird experience he was a great teacher

amazing man really little guy a really

little Middle Eastern man very little

bold very very tiny man

brilliant teacher brilliant the guy was

a brilliant mathematician brilliant I

mean just like a legendary mind so good

ah getting Goosebumps but he wouldn't

help me with my homework I was like

that's really weird why won't he help me

he seems very busy and on the test he

put the homework questions on the test

and I think they were from the book so

that's just a it's a nice way for

teachers to be able to find homework

problems so that's why a lot of times

these books

don't have answers right because the

books are chosen by the teachers right

if you're a teacher a lot of times you

get to choose the book or there's a

committee that chooses the book

especially in the advanced math classes

you know if you choose the book you want

to choose a book that you can use for

problems because you have other stuff

you have to do besides teach right and

teaching is hard you have to grade

anything that makes your job easier as a

teacher right I mean teachers are human

beings too right

so by picking these books which is a

trend a lot of books don't have answers

they can assign those problems as test

questions and homework questions so

that's another reason sometimes why a

lot of these books are so terse I think

but you do have to struggle to learn so

that's why I have so many books anyways

proof writing it's it's a struggle

um it's hard for everyone you're not

alone I've known plenty of people who've

given up on mathematics because of proof

writing

also know this one more thing before I

end this video

once you learn how to write proofs you

can get a math degree you can like that

that's the one thing you need to know

how to do because you can get through

the calculus sequence calc 1 calc 2 calc

3 d e you can be a rock star and get all

A's but then you get to proof writing

it's like uh uh so then once we want to

write proofs then you have to tackle you

know Advanced calculus abstract algebra

but you need to master the proof writing

before you can jump into those classes

you know sometimes you'll have a

statement

in an advanced math class and you have

to take like the negation of that

statement like how do you negate the

definition of a limit right how do you

do that right so you need to be able to

do that cold

how do you do a proof by contradiction

on a limit or something you know what

how do you do that you know some and

then sometimes you have more complex

definitions like you have like these

double series you know how do you deal

with that so things can get a little bit

more complicated they can get a little

bit delicate and it's in times like that

that your logic really shines those

basic core principles you learn in those

proof writing techniques come back in in

your higher level math classes so having

a good grasp of that will allow you to

get really fun you can get a math degree

you can if you can learn to write proofs

if you if you buy this book and you can

write the proofs in this book you can

you can you can do it you can get a math

degree right you can do it

but easier said than done and again the

biggest hurdle

the biggest hurdle is going to be

feedback so try to

read good proofs and again check out my

playlist on set theory it's free I've

also got a playlist on function sets and

relations it's free I've got an abstract

algebra playlist it's you know on

YouTube

and so it's got some proofs and I've got

some Advanced calculus proofs and some

topology not much but I've got some

topology stuff so

it's all correct there's no mistakes so

yeah and those proofs are pretty clean I

was always

I was always pretty good at writing

proofs that's because I had that I had

that foundational start where I took

those two classes at the same time

and I think if I wouldn't have had that

it would have been really hard like I

can't imagine if I had just taken the

proof writing class and not that

discrete math class

or or if I had taken that discrete math

class and not the proof writing class it

would have been harder for me so having

having to double courses those two

courses on proof writing really really

helped me and it was still really hard

when I got to the advanced math classes

it didn't make it easy

it just made it doable I think so yeah

what do you think what do you think

about proof writing do you have advice

for people watching this video do you

have any stories to tell about your

proof writing experience what are some

of the things you struggle with in proof

writing

do you think there's other books that

are good you think any of these books

are better than others anything anything

that can help people who are watching

this video

I hope it's been helpful as always keep

doing math