Mathematical Proof Writing
Summary
TLDREl 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
📚 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.
👨🏫 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.
📚 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.
🎓 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.
Mindmap
Keywords
💡Pruebas matemáticas
💡Estructura de las pruebas
💡Convergence test
💡Lógica
💡Clases de matemáticas
💡Feedback
💡Libros de matemáticas
💡Curso en línea
💡Estrategias de aprendizaje
💡Matemáticas avanzadas
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
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