Mathematical Labyrinths

The adventures of a post-grad math dude and current engineering grad dude.

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I appreciate the premise behind this post, but it also coerces people to stay in positions which they loathe because someone "less fortunate" would potentially "want that, too."

If your job affects your mental or physical health, you should probably leave. If your house is dilapidated or your landlord is not assisting you in fixing gas/electric/hot water or it has asbestos, linked to decreasing the IQ of its inhabitants and causing cancer, you should probably leave.

I get the premise, I do. But this is also how people coerce themselves to stay in shitty situations because they believe this shit full stop. This comes with an asterisk.

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The Myth of Sisyphus

Sisyphus is famous for his punishment in the underworld more so than for what he did in his life. The Greek myth is that, Sisyphus is condemned to roll a rock up to the top of a mountain, only to have the rock roll back down again every time he reaches the top. An eternity of futile labor as his punishment.

One story tells how Sisyphus chained the spirit of Death, so that during Death’s imprisonment, no human being died. When the gods freed Death, his first victim was Sisyphus. It is also said that Sisyphus told his wife not to offer any of the traditional burial rites when he died. When he arrived in the underworld, he complained to Hades that his wife had not observed these rites and was granted permission to return to earth to chastise her. Once granted this second lease on life, Sisyphus refused to return to the underworld, and lived to a ripe old age before returning to the underworld a second time to endure his eternal punishment.

Camus refers to Sisyphus as the archetype for the absurd hero. Both for his behavior on earth and for his punishment in the underworld. He displays scorn for the gods, a hatred of death, and a passion for life. His punishment is to endure an eternity of hopeless struggle.

What interests Camus is Sisyphus’s state of mind in that moment after the rock rolls away from him at the top of the mountain. As he heads down the mountain, briefly free from his labor, he is aware of the absurdity of his fate. His fate can only be considered tragic because he understands it and has no hope for reprieve. While at the same time, the lucidity he achieves with this understanding also places him above his fate.

The Myth of Sisyphus is what Camus calls “the absurd.” Camus claims that there is a fundamental conflict between what we want from the universe whether it be meaning, order, or reasons, and what we find in the universe: formless chaos. We will never find in life itself the meaning that we want to find. Either we will discover that meaning through a leap of faith, by placing our hopes in a God beyond this world, or we will conclude that life is meaningless. Camus opens the essay by asking if this latter conclusion that life is inherently meaningless necessarily leads one to commit suicide. If life has no meaning, does that mean life is not worth living? We are however able to form our own personal meaning and live life passionately.

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Outsider Culture & STEM: Why that kid whispering the lyrics to Vince Staples could be the next Bill Gates of his generation

The typical portrait of the engineering wunderkind has been scrawled on the minds of Americans since the 1980s. It’s usually a nerdy kid with socially awkward mannerisms, an obsessive fascination with technology and an endearing family who cultivates their interests. Try to imagine this person.

Let me guess: This nerdy kid is young, male, either Caucasian or Asian, skinny and comes from a middle-class background.

If you felt a tinge of guilt because I guessed right, let me allay your nerves: It’s okay. When I was younger, I envisioned engineers — and especially geniuses — as that, too. We’re inundated with those images when we’re young and socially engineered to continue in our current lives with those images as a backdrop. Those carefully engineered seeds are buried deep within our psyche and when they blossom, we classify ourselves based partly upon the images we see or envision. It’s one of the many reasons why many of us don’t pursue rigorous STEM fields because we simply don’t think we fit the classical definition of what it means to be part of that prestigious group.

It’s also precisely why more of us who don’t fit that classical definition mustdelve into the varied fields of STEM, not necessarily for the sake of diversity or representation, but for the sake of serving our communities — and the world at large — better than those individuals we believe are “better” at it. That fresh faced and bubbly Dominican kid whose parents own a Bodega knows the community in which he thrives better than ex-Googlers who seek to tear down the very structure which has given him a plate of food. It’s that kid who knows best how technology can help alleviate the poverty he sees or how food deserts can be eliminated. It’s that kid who knows how an app can best serve his parent’s business or even expand it.

It’s also the kid in a sweatshirt who mumbles the latest Kendrick Lamar track on the 1 train who potentially knows how to implement technology to decrease the delays at the 125th St. train station. He knows the routes, when it’s more likely to be delayed and even which train station has the most deplorable signal equipment. I can assure you that the San Francisco transplant with VC funding of $100 million won’t be the one to solve the woes that beleaguer everyday New Yorkers.

As discussed earlier, perception of oneself is oftentimes generated by the images we absorb in life. This occurs naturally and over a period of time; we’re consistently fed images of classic representations of the STEM field, so we depart from what we think we can do professionally because we simply don’t “fit the mold.” This is a terrible irony because, historically, it’s those who especially don’t “fit the mold” who succeed the most in STEM. The old school entrepreneurs of the technology sector were outcasts of society who didn’t fall in line with the masses. They’re comprised of college dropouts, flunkies who couldn’t connect with other people, irascible tempers or downright bizarre ways of thinking or behaviors. Those perceived limitations by society ultimately became responsible for what made them stand out and succeed. In short, their weaknesses became their greatest strengths. More importantly, their experiences as outsiders propelled them to greater heights than considered possible by the normative culture of their days. Outside the technology sector, the history of STEM is replete with tales of strange and eccentric personalities. Richard Feynman stands out as a representative for Physics, whereas Grigori Perelman stands out as one of particular interest in Mathematics.

Silicon Valley not only praises this history, but encourages it. Although Steve Jobs was known within the industry for his combative personality and harsh treatment of colleagues, he was heralded as the shining star of the tech industry for a time. The HBO television show, Silicon Valley, consists of a bizarre cast of characters who fit in among themselves, but act as awkward outcasts in “normal” social settings. The show has devoted itself to authenticity and hires a multitude of consultants to ensure that the hub of the tech industry is represented as accurately as possible, despite mild exaggerations of character traits.

So, why don’t we see more of that Dominican kid or Kendrick Lamar fan in STEM fields or, more precisely, in the tech sector? Simply put, the tech sector has begun to establish a specific type of outsider and actively shuns those who don’t conform to what is deemed to be an “acceptable” outsider. This is popular in group dynamics overall; once a space is occupied by a certain type of people, they actively seek to recruit others who share common traits. As such, the tech sector which originally heralded outsiders now indirectly shuns them from entering the field. This is an incredibly depressing and sad state of affairs, because it is precisely those outsiders, those outcasts of society, who can help their communities the most and contribute to the expansion of our technological world as a whole.

As a mixed Dominican/Jamaican kid born in the Bronx and raised in Harlem, I was lucky enough to have a mother who pushed me to pursue a field of academic study that brought me immense joy and satisfaction. Although I was often the only student in the classroom with that type of background, I never once allowed that feeling of “oneness” disturb me from achieving my goals. Was it difficult at times? Yes. Do I regret any of it? No, not in the slightest.

If you’re reading this and fall into the category of the “acceptable” type of outsider, please do not take this as an attack on your person hood. You must remember: You are that original outsider, that person who doesn’t fit that normative culture of society. In no way, shape or form should you feel that this is an attack on your persona, as it is not intended to be. On the other hand, if you fall into that category of otherness that’s not representative in STEM, but have a deep fascination with any STEM subject, I encourage you to follow your passion. Whether you’re a 35 year old Kansas City laborer, a 14 year old black girl with braids (or no braids!) from Brooklyn or a Harlemite with a fondness for Cuban Sandwiches (guilty), I encourage you to step forward into your passion. Your thoughts, your experiences, and your intuition will prove invaluable to the field(s). More importantly, you’ll be able to look at yourself squarely in the mirror, knowing that you didn’t allow preconceived notions or images derail you from your goals in life.

Your time is limited, so don’t waste it living someone else’s life. Don’t be trapped by dogma — which is living with the results of other people’s thinking. Don’t let the noise of others’ opinions drown out your own inner voice. And most important, have the courage to follow your heart and intuition. — Steve Jobs

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Leaving a position

I left my position as a mathematics content specialist yesterday.  Today, I give them the key and complete the offboarding procedures of the company to finalize my disengagement with them.  In addition to that, I’m planning on submitting a document which will help my former supervisor in taking over my work, although I doubt she needs much assistance from me. I have mixed feelings about this.  The contract was supposed to be reworked on January 27th, but they decided to do so on Tuesday.  I was a bit shocked by the sudden change in the contract details and I ultimately didn’t agree with it, so I made the decision to leave.  It was one of the toughest career decisions I’ve had to make, but I knew it ultimately wasn’t beneficial for what I wanted to do.  The company in question taught me immensely and I’ll take those values wherever I go, so I’m very happy about my choice to take the position overall.  That being said, certain aspects of my job were being limited, bit by bit, but it was understandable.  I was a contract worker by default and my work changed, depending on the needs of the company.  It wasn’t so much a question of money, but a question of what I thought was beneficial for my career.  I could spend the next month doing this work, being paid at a different/lower rate, knowing full well that it would stop and there would be no definitive future, or I could stop to focus on my school work and improving other facets of my life.  The decision, I reiterate, was painful to make.  I consulted my parents -- my father assisted me extensively -- and I made the heart wrenching decision to depart. Financially, I’ve been able to save up money and I feel that I’ll be ok.  I’m just saddened by the course of events, though.  Part of me is regretful of my decision to leave because I would’ve left something “undone” and I’ve only done that once in my life when I was nine years old, playing baseball on the little league team and I quit because of a temper tantrum I had.  I vowed to never do that again if it was in my power, yet here I am.  I also feel extremely guilty; I feel as if I left the company just when it needed my work the most, in a way.  That being said, I also know that my work was slowly being taken over by my supervisor.  I felt as if I was being phased out.  That played a role in my decision to leave; I felt as if my usefulness was being questioned. I also felt that I blurred the lines between contractor & employee.  This was the first time I worked under a “contract.”  Previously, I was paid as a part time or full time employee.  I would go to dinners with them, drink beer, share stories about myself and my family, open up, et cetera.  When I saw the finalized contract, I painfully realized that I really was just a contract worker.  More and more, the option to ‘’just work from home’’ kept coming up in conversation and I took it as a clue.  I felt dissatisfied with my place in work.  Ego also played a small part.  I’m at a certain point in my life where I want a bit more leeway, a bit more power over my choices; more autonomy, really.  I suppose that’s because I’m in my late 20s, so I’m used to doing things in a certain way. I don’t regret joining the company, however.  I met some extremely interesting people, engaged in some smart and dynamic work, and I had a wonderful time for the most part.  It was distressing at times and I learned a lot about myself under pressure, how I work and what’s the best environment for me.

I imagine that my immediate supervisors have negative feelings about me, though I personally wish they didn’t/don’t.  It was a painful choice to make, and a choice I’ll always have in the back of my mind as a “what if I...”  Part of me feels like I made the wrong choice; another part feels like I made the best choice ever. I’ll be a bit melancholy for a few days.  I feel as if a part of me has been taken.  Perhaps I’m taking it way too personally.  Perhaps not personally enough.

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Doing More Mathematics and Reading

Over the past few days, I’ve begun the process of rethinking how I’ve been conducting my life.  I’ve found that I’m perversely connected to my electronic devices.  This unnatural infatuation is directly tied to my obsessive commitment to work and internet addiction.  It brings me great pain to acknowledge the idea that I’m wasting precious years of my life.  I think we, as millennials, waste an absurd amount of time overindulging in online activities when we could be doing something meaningful and productive with our lives.  I once read -- online, no less -- that we give ourselves excuses why we don’t succeed the way we want to most of the time*.  I’ve often wondered about that and I know that it applies to me, just as it applies to a vast assortment of individuals I know.

I’ve always been intrigued by the technological world.  In particular, I’ve admired how tech has changed the lives of people and what we can do with it.  This is what drives my internet addiction, I believe.  When I was younger, I imagined being one of those game changers who possessed the capability of completely taking the world by storm and rearranging how people performed their daily tasks.  I could do this through technology, I thought.  When I was younger, I had a penchant of coming up with ideas and creating gadgets that would transform my ideas into reality.  If I couldn’t successfully port my idea into reality, I’d settle for a prototype that at least did something reasonable.  Somewhere along the line, however, I stopped feeding that creative spark.  Shortly thereafter, I transitioned into academia.  I don’t regret my choice to delve into academia (I’m still involved in it), but I wish I hadn’t stopped feeding my creativity.  It’s a huge part of who I am (see previous post about my bemoaning about writing), and I wished that I was a bit more forceful in implementing it in an academic atmosphere.

That’s all done with, though.  The past is the past and we must move forward if we are to succeed, lest we stay shackled by regrets.  One of the things I’ve been doing is reading more.  As you are all aware, I’m reading through Mathematical Mindsets by Jo Boaler and, before I jumped onto that book, I read a martial arts autobiography by Judd Reid, who is a Kyokushin Karateka.  Aside from Boaler, I’m also beginning to trudge through Thomas Piketty’s celebrated, “Capitalism in the 21st Century.”  It’s a massive book, weighing 3.2 ounces and totaling over 696 pages of massive information.  The style of writing displays a mixture of academic rigor and standard prose.  The book was written in a manner to welcome non-economists and economists alike, but continues to maintain standards commonly found in academic papers.  It holds a place right smack in the middle; it’s an admirable feat.  For one to fully absorb the book, they’ll need to carefully absorb each word and read a paragraph a few times to really understand what’s truly being said.  It’ll take some background research and reading if you’re fully invested in what the book has to offer, I believe.  Once I delve into it a bit more, I’ll be able to fully expound on my thoughts regarding the contents of the book.

I’ve also made the decision to do more mathematics.  That’s why I am restarting, as some of you may have missed, my set theory primer series. I’ve missed doing mathematics on my own time and I plan on brushing off the cobwebs.  I’ll also be starting a dynamical systems primer.  I’ve seen a dearth of information regarding dynamical systems and I’d like to create a set of online notes that could double as a starting point for any budding mathematician and as a refresher for math majors who have taken courses in the field.  I’ve taken several courses in dynamical systems (Linear Dynamical Systems, Nonlinear Dynamical Systems, Linear Control Theory, Robotics Simulation & Design, Advanced Dynamics, Dynamical Systems), and it’s what my MechE PhD is mainly focused upon.  In this primer, I’ll introduce the primary concepts and ideas, then solve theorems and a multitude of problems.  For this, I’ll most likely use a combination of notes I have acquired over the years and a hodgepodge of notes from several dynamical systems books.

Thanks, once more, for reading an excessively long post.  Hope to update you all soon.

*Excluding those who have a multitude of commitments, such as family care, extra work/school responsibilities, etc.

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Some new changes and a new perspective

After receiving the last comment on my blog, I decided to do some research into Tumblr themes and added a Like/Reblog button to the bottom of my posts.  They don’t look especially pleasing to the eye, but it will allow people to reblog/like my posts with more ease.  I’m also planning on changing the theme of the blog or relearning HTML/CSS to create an entirely new theme, one more in tune with my personality and love of mathematics.

In other news, I’ve recently begun reading this book “Mathematical Mindsets” by author Jo Boaler.  The book was given to me by the CEO of the Startup I work for as a gift of sorts, and it’s been incredibly enlightening.  The book is about how mathematics is taught, the two different mindsets it can produce (fixed & growth) and how we can transform our students from being learners with a fixed mindset to learners with a growth mindset (these terms will be defined shortly).  The book also discusses ways in which we can improve mathematical education, by changing the way in which we view and teach mathematics.  It’s been incredibly rewarding for me as a PhD student to read this book and finally comprehend why I think the way I do, and why I sometimes struggled with academia.

The author, Boaler, posits that students have two types of mindsets: Fixed mindset and growth mindset.  A student with a fixed mindset is under the belief system that intellect is innate and, despite best efforts, cannot grow.  They are immediately prone to believe that if they get a math question incorrect, it means that they’re either stupid or not a “math person.”  This has unspeakable effects on the self esteem and limits their career options tremendously if they yearn to pursue technical fields requiring mathematics.  The fixed mindset belief is widely espoused within STEM among students and teachers/professors alike, creating an environment of high stress and depression.  Although I never suffered from overly debilitating depression, there were times in which I suffered from feeling profoundly foolish due to my belief that I had an inability to retain, learn or even apply a mathematical concept (I later overcame any mathematical issue through studying, perseverance and hard work).  I suffered from a fixed mindset myself for years, until I started questioning the idea of intelligence a few months ago.

The growth mindset, on the other hand, teaches us that intelligence grows over time through trial, effort and learning from our mistakes.  It’s tied to the idea that hard work and persistence pays off toward accumulating skill and intellect, not the idea that there is an unchanging version of intelligence inside of you that cannot be changed.  It’s this mindset, Boaler proves, that tends to be the most useful for students.  Students with this mindset are oftentimes able to learn mathematics in a more intuitive manner and capable of making striking relationships between mathematical concepts.  Patterns also come far easier to them in this way.  She found that students with a growth mindset score higher than those with fixed mindsets and think mathematically, rather than memorize random facts to assist them.  Number sense, for example, is a way of thinking mathematically.

In the book, mistakes are often highlighted as a method which gives students a chance to learn more.  In fact, if a student makes a mistake, their brain develops a synapse.  Yes, that’s right -- the brain actually grows when a mistake is made!  What is more surprising is that this growth occurs regardless if the student has a fixed mindset or growth mindset!  Those with growth mindsets, however, have a clearer sense of the mistake they make and are quick to learn from it, whereas those with a fixed mindset have a harder time in doing so.  This difference is staggering: brain activity is more prevalent in those with growth mindsets than those with fixed mindsets.  This also leads one to suspect that the fixed mindset follows them throughout life, outside of mathematics, and into their daily lives.  It could severely hamper how they treat interpersonal relationships, professional relationships and even their choice of work.

Personally, this book was a revelation to me.  When I was younger, I had a fixed mindset of sorts.  My teachers always told me that I was smart, so I felt great when I was able to get something right (you’re so smart!) versus when I got something wrong (oh, I thought you were smart?).  Eventually, as I transitioned into junior high school, I changed the way I viewed myself.  I started to look at myself as intelligent/smart, but someone who also had to work quite hard to achieve the results I wanted.  It’s interesting to note that I had started to read philosophy quite a bit during this time.  I began to transition into the growth mindset until I reached HS, where I met people who would castigate me if I got something wrong or praise me when I got something right (this ultra-perfectionist culture is also touched upon in the book, I should mention, and Boaler is of the belief that it ultimately damages students in the long run).  Eventually, these people began castigating me even when I was correct just to damage my self esteem.  I’m afraid to say that it worked to a small degree and to this day, I still continue to question my worth in mathematics from time to time.

I’ve always been someone who has looked at mathematics from a very distinctive point of view and this book, surprisingly, confirms that my point of view fits that of a mathematically inclined person.  Shockingly, however, academia does not implement this growth mindset and it is especially strong in STEM.  Most professors and students alike strongly hold onto the conviction that your ability to be good in STEM is innate.  They (and even I did, from time to time) laughed at those liberal art majors who simply couldn’t handle the rigors of being a STEM major because it was too hard for their brains.  This pervasive idea backfires even on us students. however.  When we receive a bad grade on an exam or a class, we immediately consider leaving the major (how many of you STEM grads had this thought at least once or twice a year, and seriously considered it? be honest).  You are also punished psychologically for making a mistake on a mathematics/STEM exam.  Immediately, you are implicitly told that you are not good enough for the subject and to move on.  I’ve even head a professor tell a student that maybe he’s just not inclined to mechanical engineering and should leave the program (he didn’t and was was able to land on the Dean’s List in his final two years of undergrad).

I’ve long held doubts about myself, my abilities and, worse yet, my ability to learn.  I’ve always been under the belief that I’m going to irrationally lose some sense of my capability to do.  This has always been my wildest fear, really: not being able.  I recognize that this is a trait from someone with a fixed mindset rather than a growth one, though.

I encourage you all to read the book if you get the chance.  The book also offers some really cool math puzzles and games!  One of them I tried out was, “Is it possible to derive the numbers 1-20 from only four 4′s with any operation?”  That was a fun game.

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ive recently come across your blog and have read all of the posts you have written and found them very intriguing, however its very difficult to follow or like your posts as there arent any available buttons to instantly press. i think the only way people are able to like or reblog your stuff is if it happens to come up on their homepage. It might just be my computer playing up but i thought it best to mention as you may be missing out on viewers

Thanks for the info!  I never knew that was a potential issue until now!  I appreciate you pointing this out.

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On the nature of reading & writing.

Long Post Ahead!  Read at your own risk!

When I was younger, I used to write consistently and almost incessantly.  It was a personal hobby of mine, one that I enjoyed immensely and one that I cherished.  I didn’t write to necessarily enhance my craftsmanship capability, but I did become fond of whatever skill I acquired during my long excursions of writing.  As a homeschooled child with not much social interaction, writing became pivotal to both my psychological and academic growth.  Through online communities, I was able to explore my writing capabilities further by delving into worlds unbeknownst to the world around me.  In addition, I found individuals like myself who enjoyed writing as much as I did and developed great relationships with some of those people.  Some of those relationships, in fact, are still in tact and I keep in contact with them.

Somewhere along the line, however, I lost touch with writing.  By losing touch with writing, I lost touch with a vital part of who I am and who I aspire to be.

When I enrolled in University, I continued to write.  As my academic responsibilities increased, however, the decision to pick up my pen and engage my creativity or study for a test became less and less of a struggle for me; I’d always just study.  After the second year, there was no decision to be made, really -- I would simply pick up my book and study.  This was a natural progression, I thought.  We inevitably leave behind certain hobbies we develop as children or as teenagers, and become responsible, mature adults who cannot be bothered to engage in such trivial matters.  Although I never openly admitted this to others or myself in such blatant terms, I believed it to a large degree.  By the time I graduated with my B.S. in Mathematics, I hadn’t done any intensive writing aside from the occasional papers I had to write for my liberal arts/history classes.

Funnily enough, however, University introduced me to a wide variety of books on an eclectic mix of topics.  Although my library was quite expansive, the University setting truly brought enlightenment in regards to certain topics.  Being homeschooled by a single mother who worked and went to school herself, she gave me a listing of books the public schools used, instructed me to read them and write papers about them.  I did as I was told, and she graded them appropriately (she had a B.A. in English Literature, so she knew what she was doing).  That being said, I was never exposed to the niche field of HS books that most kids pass around when they enter their phase of self discovery.  A friend of mine from the other side of the United States, Hima, introduced me to The Stranger by Camus.  We were both quite melancholic individuals, distraught by the realities of life, and we both found solace in existentialism.  Before University, The Stranger was the only book I had read which covered existentialism and absurdism to a degree of immense depth.  I’ll forever be thankful to her for introducing me to this and we still maintain contact with one another, though our discussions are quite sparse due to our different career fields; she is a Doctor and I am a PhD student/mathematician/education specialist.

In the second year of my undergraduate degree, after I had practically abandoned writing, I took a Philosophy class on the idea of Friendship.  I had not necessarily been too invested in the class at first; I only chose it so that I could fulfill my liberal arts requirements and focus mostly on my math courses.  Needless to say, this was quite possibly one of the most fundamental college courses I ever had the pleasure of taking.  Although the course description was about exploring the idea of friendship through antiquity and current times, the professor gave us book recommendations which expanded philosophically on love, interpersonal relationships and how we view ourselves through said relationships.  Through this course, I was introduced to Nicomachean Ethics by Aristotle, Michel de Montaigne’s Essays and St. Augustine’s Confessions.  While other students read only what was required of them in a cursory fashion, I devoured these books.  These books, especially Nicomachean Ethics, instilled in me a hunger for reading that I never shook.  The next semester, I signed up for two more literature courses for the sake of the coursework alone.  My increased workload made it practically impossible for me to sit down and write, but I figured I could read.  By signing up for these courses, I had psychologically “tricked” myself into reading more!  If I was signed up for these courses, then my goal was to obtain an A.  For me to obtain that A, I needed to know the material inside-and-out.  Therefore, I had to read more than ever.  I also had to write excellent papers to receive good grades.  Although I wasn’t writing from my heart as I did once before, I was at least writing more than I had in the previous year and a half, so I was joyful about that.  Nonetheless, reading became my “new” personal hobby that I could explore without feeling guilty if I didn’t pick up a math textbook and study.

By the final year of my undergraduate education, I had been exposed to several in depth ideas about existentialism, absurdism and classical literature that I could’ve been a double major in Mathematics and Literature.  I chose not to travel down this route, however, as it would take another semester for me to graduate.  Instead, I took this newfound knowledge with me and entered into Graduate School.

Graduate School was difficult for me to handle in the beginning, on a personal level.  I couldn’t read the literature I wanted to, as I needed that time to engage myself with extremely high level mathematics and study independently.  I graduated within two years with my M.S. in Mathematics, but I had only read two books as a leisurely activity.  It was saddening to recognize that my secondary and deeply personal hobby, reading, was now taking a backseat to my deeply personal and professional relationship with mathematics.  I don’t have any regrets, but I do wish I had managed my time more efficiently so I could balance the two more.  I did think, however, that maybe this was simply just another natural progression.

Upon graduating with my M.S., I went into the workforce as a Coordinator/Manager of an after school program.  I was their Mathematics Instructor and had built a strong history with the program over the years.  Professionally, I did not envision this for myself, but I enjoyed interacting with teenagers and young adults.  The salary I earned didn’t quite match my credentials on paper, but I was somewhat content with it; it didn’t keep me from pursuing other goals and it didn’t restrict me.  If you’re a working man/woman reading this, you’ll know that FT positions rarely give you the time to sit down and pursue a side hobby unless you truly dedicate time to it and manage your time effectively.  I was still slacking in that regard, so I couldn’t/didn’t pursue my hobbies as much as I should have.  I did, on the other hand, delve into my long forgotten hobby of playing Chess, which was a great joy for me to reestablish once more.

I eventually left the position due to my yearning for growth, and a host of other reasons, and I accepted a slot as a PhD student at a well respected university in the field of Mechanical Engineering.  I had looked forward to simply being a student without “working,” but I had perused the “Open Jobs” section that my University has for current students/alumni and saw a position as a mathematician/mathematics content writer.  I sent my resume because it looked interesting and, wildly enough, I was contacted.  I went through the interview process, signed the necessary paperwork and before I knew it, I was hired.

The company is a tech start-up company and I’m their in-house mathematics expert who writes their math content.  It’s an intensely collaborative environment and has sparked my yearning to practice writing code again, as most of my colleagues do nothing but write code to translate the mathematics content I’m writing into an easily accessible format.  More than that, however, everyone places health at the top of their list.  Mental and physical fitness are important concepts to them individually, and although I’ve never placed much credence on fitness in the past, I’ve slowly begun taking better care of myself.

One of the people I work with has an intense engagement with photography as a hobby.  I also used to work with another hobbyist photographer. It is a relatively personal and experimental journey, and I’ve had the pleasure of viewing some of their photographs, but it made me realize something: It is possible to maintain your professional goal and continue to enhance yourself, spiritually and otherwise, through other personal hobbies/endeavors.  For so long, I had been under the false idea that one must fall under labels.  I am a math major, therefore I only do mathematics and nothing else; I am a writer, therefore I only write, ad nauseum.  I realized, upon reviewing some of my coworkers' photographs and speaking with others, that I’m not wrong for wanting to read more, for wanting to write, for wanting to explore.  I created this blog two years ago in an attempt to actually write more, even if it was about mathematics.  My yearning to express myself has been pushed down for so long that I sometimes talk incessantly.  I’m a relatively private person, but even people who are private need an outlet to express themselves.  For so long, I have fooled myself that I only need academics or work to express myself.  Looking back, I realize how ridiculous that notion is.

This is an excessively long post, quite possibly the longest post I’ve ever had the pleasure of writing.  It was necessary, however.  It was necessary for me to write this much and to convey my feelings in this manner.  I doubt most people will get to the bottom of this post, but it has lifted a weight off of my chest.  I am joyful for the opportunity to write and pursue this passion of mine once again.

I’ve said this over and over again on this blog, but I will genuinely try to dedicate more time to this blog.  I’d like to write more about mathematics history, mathematics tutorials and, more importantly, myself.

Thank you for listening.

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Back & on track!

Hello, all. I most likely have no active followers anymore, but I thought I’d update this Tumblr regardless of that fact.  So, let’s clear up some information.

Why were you on hiatus?  What happened? I was caught up in working plus going to school full-time.  In addition, I stopped having enough time to blog/have fun on the internet.  I also felt somewhat melancholy over my potential career prospects, and decided to stop blogging on mathematics to see where I wanted to take my life.  Now that I’m a bit more stabilized, much more upbeat and feeling better about where I am in life, and where I want to be headed, I figured I’d start blogging again. What are you doing now? I’ve been recently accepted into a Ph.D. program in Mechanical Engineering, focusing on dynamical systems & control theory.  During the last year of my MS in Mathematics, I took advanced graduate coursework in dynamical systems and enjoyed it immensely.  I also became enamored by the field of Mechatronics and the prospect of actually applying my mathematical knowledge to a growing and diverse engineering discipline.  In short, I stopped feeling regretful as I once did.  Classes start soon and I’m certain that I will be contributing to the blog a lot more to discuss problems, propose solutions and to divulge more cool information. So, does this mean no more math?  Only engineering? My heart will always be that of a mathematician, but my hands will be working toward the goal of engineering.  In other words, my profession will be engineering, but my first love will be mathematics.  I plan on continuing my set theory series, discussing interesting (read: undergrad/graduate) mathematics problems and proposing nice, doable problems.  I also plan on integrating engineering concepts and problems.  You may find me uploading Raspberry Pi/Arduino/Intel Edison videos, personal/professional projects,

More than anything, I want to expand this blog as my personal project and lead all of you into the intriguing facets of my life.  Moreover, I’d like this blog to become a pathway for budding young mathematicians and engineers -- from teenagers to adults.  I was confused most of my professional/academic life and I hope I can alleviate some of the anxiety or fears from others.

I’ll be updating this blog again shortly (within the week), so stay tuned!  I have to earn the trust of my followers once again, and gain some new ones.

Until next post -- ciao!

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The brutal truth for Math majors

Sadly, due to a multitude of responsibilities in my personal and professional life, I haven’t been able to update this blog as often as I would like.  Nevertheless, I’ve some time now and I’d like to write a sizable and detailed piece on a topic that’s not only quite important, but useful for those who are majoring in mathematics currently or planning on attending graduate school, specializing in pure mathematics.  When I say pure mathematics, of course, I mean sub-fields of mathematics in which there exists no inherent or immediate application to the real world.  These fields include, but are not limited to, the following: Mathematical Logic, Algebraic Topology, Set Theory, Geometric Group Theory, etc.  There are many more examples of pure mathematics, but those aforementioned examples should suffice for the reader.  This post, I hope, will illuminate the minds of readers, mathematicians and non-mathematicians alike, to the brutal truths surrounding the field of mathematics and the job market.

Let me be blunt here: This will likely make many math majors feel sick in their stomach or break their hearts, but I’d rather break it to you bluntly than allow you to coast through your undergraduate – or graduate – years with blinders on.  If you’ve made the decision to major in mathematics only, strictly mathematics, without any minor in anything applicable(i.e., engineering, computer science, finance/economics, etc.), you’re setting yourself up for a difficult road ahead in your future career.  Mathematicians and physicists share one thing in common: We are both extremely prideful individuals who allow pride to cloud or enshroud our better judgment, as ironic as that might seem.  The fact of the matter is that, although we brag about the supposed purity of our field, we delude ourselves into thinking that we are somehow above the trivial pursuits of society.  The fact of the matter is that most of us are simply not.

As human beings, we have evolved significantly.  We started out as mere single celled organisms and grew into rational beings, capable of building lavish kingdoms and governmental structures which adhere to a set of rules.  We’ve been able to progress this far because, throughout history, there have been individuals who made the decision to forsake capitalistic gain for enlightenment and spread the knowledge they acquired to the mass populace.  Although the fields of science and technology are extremely useful and applicable now to the real world, it all started with certain individuals posing one simple question: how does it work?  Asking this alone was not only celebrated, but encouraged.  We benefited, both directly and indirectly, from a multitude of individuals celebrating this because it meant that humanity could move one step forward at a time.

It’s time to face the facts: We’re simply not like this anymore.  We are now primarily judged on what we do, not on what we ask.  It is not enough to be inquisitive and to learn, but to apply what we’ve learned to the society we have now.  This isn’t inherently good or bad; it’s always been like this to a small degree.  It’s simply on a larger scale now more than ever.

There is a taboo in Academia that no one wants to discuss, especially current and future mathematicians, and that is the nature of tenure.  Professorship opportunities at very good universities are scarce and, naturally, current and future mathematicians want to secure these positions within the top universities because they guarantee better career and funding opportunities.  In addition, being surrounded by like minded individuals who share your research interests will simply bolster your knowledge; being around them will force you to be the very best you can be and you can only become the best when you’re surrounded by fellow mathematicians who are willing and able.  We envision being within the Harvard Mathematics department not only because of the prestige attached to the name, but also because of the opportunities presented there.  Due to the competitive nature of mathematics and the fact that some are “naturally” better than others, all of us won’t reach the apex that is the Harvard Mathematics department.  Nevertheless, we don’t cry too much when we have to write Princeton or Courant on our resumes(Just kidding, Princeton/Courant!).

It is far more likely that a mathematician, armed with a PhD from a well respected university, will attain a professorship at a university that is well below his/her academic educational level immediately after graduation or jump into a post doctoral position at a very well respected university, earning a meager wage.  I have yet to see a mathematician, aside from a select few from the top ten universities of the world, join as a tenure-track professor at the same university or similar one.  Normally, new PhDs enter post doc hell or simply make a few steps down in order to secure employment.  The latter option isn’t inherently horrific or terrible, but I doubt that is what many mathematicians envisioned for themselves when they look at the their Yale University degree.

Pure mathematicians are in a dire predicament.  Shunning applicability and brimming with pride after reading G.H. Hardy’s A Mathematician’s Apology, they strive to bury themselves between the dense pages of Hatcher’s texts, Rudin’s proofs and Herstein’s expansive exercises.  Some delve further, attempting to study the impossibility that is Bourbaki.  Regardless of our independent levels, we are naturally bred for an academic environment and we have willingly accepted the duty to become scholastic monks filled with the idea that our studiousness will, one day, pay off in dividends within academia.  We study mathematics for the sake of mathematics, not because we find an inherent financial gain to it.  We are told, both implicitly and explicitly, that there is an enormous financial gain attached to our impressive mathematics major and we coast through our undergraduate days with this idea in the back of our head, but we don’t necessarily think about it.  We just do math because it’s fun, period; the financial gain is a subconscious thought, though a very important one.

Like everything, however, there exists fine print.  In order for one to make the mathematics major “useful,” one must consider the actuarial/financial/business/software engineering track.  As such, it is important to load up on courses which depend on applicability and learn how to apply what you’ve learned in companies where you’re judged solely by your ability to produce.  In essence, you’re discarding absolutely every word G.H. Hardy wrote and selling out.  Some of us choose to sell out and become the embodiment of everything we consider wrong with mathematics; others choose to live and -- metaphorically -- die by our honor code.

And die, you shall.

On one hand, you’re bred for academic career.  Academic institutions should, by all accounts, love you.  Assuming that you’ve done reasonably well in your courses and exams, you’re guaranteed to obtain a spot in graduate school.  Another three to four years in academia is no biggie: you’ll learn new mathematics and, if all goes well, produce new mathematics.  It’s an exciting time.  Upon leaving graduate school, you should be able to find employment at a university where you’ll be able to continue your research endeavors and teach your passion, which is mathematics.

On the other hand, you’ve essentially made yourself into a one-trick-pony.  Pure mathematics is great, the multi-millionaire says, but what does it matter if you cannot apply what you’ve learned in the real world?  Furthermore, what about profit and basic self sufficiency?  Group theory is an exquisite field of study, but you can’t explain the definition of a group to your landlord when the rent is due or when your wife/husband is ill.

Many theoretical mathematicians and physicists are learning this the hard way and, angry at the ghost of G.H. Hardy, are disavowing their previous theoretical leanings for a more applicable field of study.  Many are fleeing academia entirely due to the bleak prospects of employment, and running into fields where there exists steady and guaranteed employment. These fields have been mentioned earlier, so I shall not run the risk of repeating myself. Nevertheless, pure mathematicians are slowly succumbing to the realization that they have no immediate applicable skills to offer to the world at large besides prove. It is not uncommon for many pure mathematicians, armed with a BS/MS, to attend graduate school to study an intensely related, but fundamentally different, subject matter. Even I have considered the actuarial pathway and, to be completely honest, I am still considering that particular pathway. The purpose behind this post is not to shatter the hearts of would-be mathematicians, but to prepare them. If you choose to major in mathematics, strongly consider double majoring or minoring in an applicable academic subject, such as computer science, financial/actuarial mathematics, statistics or an engineering discipline. I've never regretted my decision to major in mathematics and I still don't regret it, despite the gloomy nature of this piece. I simply wish someone, caring about my financial future, spoke with me in depth about the decision I made. Truthfully, my alma mater invited many actuaries and former mathematicians-turned-financial analysts to discuss opportunities within the financial industry. That being said, I never necessarily found myself interested in those fields and merely yearned to study and do mathematics. When asked by fellow students about my potential career field, I gave shallow answers that alluded to studies which rank "mathematics" as the top major for potential careers. Others before me said this and many more after me will parrot these "studies" or "articles," which never get to the root of the primary type of mathematics one should focus upon.

Do yourself a genuine favor and take this advice.  Continue your pathway toward enlightenment, but do so with a possible career field in mind.  Do not engage in egotistical debates or rants with fellow like minded mathematicians and physicists about the purity of your field; it simply does not matter in the long run.

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Fluid Dynamics & Measure Theory posts incoming soon.

This semester, I'm currently taking graduate courses in Fluid Dynamics and Measure Theory.  So, if all goes according to plan, you will all be learning fluid dynamics & measure theory with me since I'd like to get into the habit of tabulating notes online.  I'm also sitting in on a graduate level PDE class, so I may sprinkle in some of that this semester.  I'll return to my set theory primers sooner or later, but I'd really like to discuss the mathematics as it pertains to my educational goals this semester. Hopefully, this tumblr will become a quasi-repository for a voluminous amount of problems found in both measure theory and fluid dynamics.  Mathematics is definitely difficult, but what makes it extremely frustrating is the lack of help and guidance in the upper echelons of the field.  During undergraduate senior year and especially during graduate school, you are more or less expected to figure everything out on your own with minimal help/assistance from the professor(s).  Although classmates are sometimes willing to help, it's usually considered best for you to do about 99.9% of the grunt work -- which I genuinely don't mind.  It just so happens, however, that the grunt work, while intellectually rewarding, can be allayed somewhat if someone was there to simply point one in the right direction toward the solution.  Hopefully, this blog will help some people who are struggling with either MT/FD. I plan on publishing solutions to problems found in books, as well as proofs.  Currently, I'm using these text books: - Measure Theory, Royden, Real Analysis.  2nd/3rd Edition.  I switch between them. - Fluid Dynamics. Childress, An Introduction to Theoretical Fluid Dynamics. Acheson, Elementary Fluid Dynamics. Remember, my policy regarding solutions/proofs is pretty open.  If you believe you've spotted an error in my reasoning, please feel free to message me!  I'll gladly correct it and I will, of course, give credit where credit is due.  I strongly advise, however, that one not completely copy a proof that I write up, simply because it'll affect your learning process, not mine.  In addition, if I make a subtle error in a proof and you copy it, you're the one who will get the bad grade from the professor when/if you hand it in! Anyway, I've got to go.  Tomorrow, I should be able to write up two or three posts.

P.S., I'm a bit bitter that my review received no replies or notes. ;(  Oh well!

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Well I see you like maths, do you enjoy physics too? :)

Yes.  In particular, I enjoy the application of physics in Mechanical Engineering.  If I had to do it all over again, I'd most likely major in Mathematics with a minor in Mech E.

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Story Review and analysis: The Metamorphosis by Franz Kafka.

It's been a while since I've posted, hasn't it?!  I've been quite busy due to the constraints of school and working.  Despite my recent hiatus, I'm here to resume my set theory primers and delve into other facets of mathematics that I may find intriguing.  Fear not, I shall resume my set theory escapades soon; it just so happens that this post isn't dedicated to mathematics in the slightest! The title more or less gives it away: This is a review of the story, The Metamorphosis, by Franz Kafka.  Although I find my head buried in between the notes of formulas and proofs more than half of the time during the day, I attempt to put away a portion of time throughout the week to read leisurely.  Recently, I've made the decision to reread some stories I've read in my youth and discover new ones.  Being older and a bit wiser -- not too much, I'm afraid -- I find that I'm able to understand certain facets of stories a bit more clearly and find parallels that I thought never existed before.  To be quite honest with you, these reviews will be more personal rather than objective and, as the reader, you should take my reviews with a grain of salt.  That being said, I hope that you'll be able to glean something useful from my reviews and enjoy them. Now, as with any review, there must be a ranking system or a judgment criteria.  In an attempt to appease my (probably gone) mathematics audience, I'll rate each book/story out of five.  Zero will be considered absolute garbage, whereas five will be considered a masterpiece.  Naturally, fives and zeroes will be extraordinarily rare, but I'll assign those numbers if the book/story is just that terrible/incredible.  I may incur some disdain for some of my opinions, but that's the beauty of living in a democracy: Everyone is entitled to their opinions and, more importantly, open debate never hurt anyone!  Well, civil debates anyway. Let's begin! The Metamorphosis by Franz Kafka - Review.

Kafka plunges us right into the absurdity of the story by beginning with the following sentence: "As Gregor Samsa awoke from unsettling dreams one morning, he found himself transformed in his bed into a monstrous vermin." The sentence, whilst quite simplistic in prose, immediately garners the attention of the reader.  This introduction is one of the more infamous ones in history, along side Camus' introduction from L'Etranger(Maman died today) and it discreetly cautions the reader to suspend their sense of realism in favor of the surreal. Naturally, it is expected that the protagonist, Gregor Samsa, will fall into a frenzy of panic and worry while simultaneously attempting to understand his current debacle.  Instead, Samsa simply regards his recent transformation as a mere inconvenience and decides to slumber once more.  Perhaps this is nothing more than a terrible dream.  This plan, if you can even call it such, is torn asunder once Samsa realizes that he is unable to sleep on his side.  As a bug, Samsa discovers, he is unable to lay on his right side -- his favorite side -- due to his newly acquired physiology.  Seemingly both oblivious and accepting of his fate, his mind wanders and he begins to think about the perils of being a traveling salesman.  Finally, the crescendo of this entire incident impacts him like a grenade: He will be late to work. Cleverly and immediately, Kafka constructs a world in which the absurd displaces the norm and if one is inclined to survive in this Kafkaesque reality, they must submit and abide by his rules.  As the story progresses and the reader continues, we see how Kafka is able to bend his own reality and possibly even shatter it.  Although we may accept this Kafkaesque reality, this does not -- and should not -- imply that the other characters will.

This newly designed world also includes other inhabitants, which mainly consists of his family members.  Although they live in this Kafkaesque world, we witness, through their treatment of Samsa, their unwillingness to accept it.  Although they openly reject such a world, they also paradoxically live within it or, more directly, beside it.  They react the way in which many of us in this world would react to a man-turned-vermin: disgusted, confused, violent.  The characters, while somewhat exaggerated to a small degree, exemplify our reality.  They comprehend the fact that they are absolutely useless in this "world," but they would rather choose to accept their futility than embrace it.  In lieu of fighting against that which they cannot, they make the choice to simply ignore Kafka's reality.  In the story, this is shown through their outright disdain and dismissal of Samsa.  Cast aside, Samsa slowly comes to the full realization of his condition and he can't help but accept his fate with both regret and sadness. This is where the story truly takes off and where we make the choice to privately reject our reality in favor of this Kafka's. Progressing through the story, I found that I unashamedly empathized with Samsa and I even yearned to assist him in his current state.  Paradoxically, I regarded the other characters as interlopers, as enemies.  As representatives of my world, I should feel a deeper connection with them.  Happily, however, I did not.  I openly pushed aside my own reality and gleefully accepted the reality in which Kafka had created for Samsa.  By reality's standards, I'd be labeled a lunatic at worst; mentally disturbed, at best.  Regardless, I flung myself entirely into this world and I wept alongside Samsa as the story neared its inevitable conclusion. Kafka's "The Metamorphosis" coerced me to ponder several possible philosophical parallels.  It is abundantly clear -- at least to me -- that this story has entrenched ties with absurdist philosophy.  As the reader, we accept the absurdity of Samsa's situation -- much as we accept the absurdity in the world -- and we make two choices, given our predicament: acceptance of this absurdity, or rejection.  The humanistic characters within the story represent our natural inclination to reject the absurd, to turn a blind eye to that which is, at times, staring at us right in our face.  Our willingness to accept Samsa's reality represents Man recognizing the absurdity of the world and living within it, building within it.  Ultimately, Kafka is posing a philosophical question to the reader at the end of the story. Some reviewers and literary critics are quick to discount Kafka's short stories as making no sense, and they claim that is precisely the point of many -- if not all -- of his short stories.  I find fault with this reasoning, especially if you're able to extend his stories from the literal and conceive of the possible allegorical concepts Kafka was attempting to convey.  Of course, not all of his work is meaningful or philosophical; however, simply repeating the motions of his work without a thorough analysis is insulting, both to his memory and his everlasting effect on the field of literature. Rating: 4.8/5.0, Short Stories Category. Protip: I'm running on no sleep at all!  Excuse any spelling errors or grammatical mistakes, in addition to some awkward phrasing.

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I see a lot of people on tumblr talking about how screwed up it is that we force kids to take math up until Algebra II, and most of them won’t need that much.

But no one talks about how we all have to take 12 years of English and 12 years of history. Those weren’t my best...

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While I mildly concur with your assessment, OP, I dislike your subtle disrespectful tone against literature and history.  Both of those aforementioned subjects help the intellectual growth of an individual.  I was fortunate enough to be naturally inclined to literature, history and mathematics; therefore, I've never felt an aversion to either subject.  That being said, I understand other students are not like me and they possibly feel angered when they have to learn a multitude of other subjects which leave a sour taste on their palette. I understand that you feel passionate about mathematics -- at least, that is the what I'm deriving from your original post -- but please do not indirectly disrespect those other fields of study.  It shows that you're at the same level of the individuals you're attempting to disprove: you're willing to disrespect/bully another academic discipline(s) because you feel ''pride'' for your own. If you really feel pride for mathematics and want to defend it against naysayers, I suggest doing so on a higher ground.  Stooping to their level will only diminish your integrity. Just for future thought.

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Basic introduction into Set Theory, part III.

I know this has been a long time coming, but alas, I've some time to kill on the computer.  This one, as you can expect from the other posts, will be quite long and will contain a slew of definitions.  Let's get started then, shall we? In our last post, we defined a union of two(or more) sets, an intersection of two(or more) sets, a finite/infinite union/intersection of many sets and also introduced a symbolic definition.  We also went over an important example which essentially built up \( \mathbb{N} \) using strictly our definitions.  Now, let us delve further into set theory! Definition 1.6.  Let \( F \) be a non-empty set such that each member of \( F \) is itself a set.  In other words, suppose \( A_{1}, A_{2}, A_{3} .... A_{n} \in F \).  Then, we call \( F \) a family of sets. Example. A good example -- a very good one, in fact -- is the power set.  Let's define it now! Definition 1.7.  Let \( S \) be a non-empty set.  We say the power set of \( S \), defined as \( P(S) \) is a family of \( S \), such that it contains each subset of \( S\) as a member. Ex. Let \( S \) = {1, 2, 3}  Then, \( P(S) \) = { {1}, {2}, {3}, {1,2}, {1, 3}, {2, 3}, {1, 2, 3}, \( \emptyset \) }. Let's explain some of the members of the power set before we move forward to explain what that squiggly deformed circle is doing in there.  For those of you who are unaware, that 'squiggly deformed circle' is the empty set. Clearly, {1}, {2} and {3} are subsets of S.  Definition 1.1 is not disobeyed at all here.  Each set contains an element which is a member of the 'parent' set \( S \).  The same logic follows when we look at {1, 2}, {1, 3}, {2, 3} and {1, 2, 3}.  Now, let us discuss the foreign symbol which is listed last within the power set.  First, what is it? The empty set. The empty set is one of the most trivial sets in set theory, and yet it continues to be a very dangerous set because most amateur mathematicians make the mistake of underestimating it or not accounting for it in certain proofs in which it is necessary that the empty set be included or excluded.  First, let us define what the empty set is, although the name should be slightly obvious. Definition 1.8.  Suppose there is a set such that no element exists within the set.  We define this set to be the empty set.  Symbolically, we write the empty set as { } or, more commonly, \( \emptyset \). Generally, the empty set acts as an elusive concept to those who are quite foreign to set theory and mathematics.  Many who read the definition feel that they comprehend it, but it's only after a few problems do they feel somewhat overwhelmed by their true lack of understanding.  Let's discuss this a bit more in depth. Mathematically speaking -- as if we are discussing anything other than mathematics -- the empty set is not 'nothingness.'  It is not a set which is just there for the sake of being there; it is a set which simply does not contain elements.  Furthermore, the empty set is unique under any context.  Thus, if we are discussing the empty set in \( \mathbb{N} \), we are also indirectly discussing the empty set in \( \mathbb{C} \).  Let's also go through some thought provoking examples and prove certain properties associated with the empty set. Example 1. What is {\( \emptyset \)} \( \cap \; \emptyset \)? Solution. \( \emptyset \). Example 2. What is {\( \emptyset \)} \( \cup \; \emptyset \)? Solution. {\( \emptyset \)}. Note the dichotomy here.  This is very important. Example 3. What is {\( \emptyset\)} \( \cap \) {\( \emptyset \)}? Solution. {\( \emptyset \)}.  If you answered those questions correctly, you've got a good grasp on the concept of the empty set.  If not, please review the definition again or simply send me a private message.  I'd be happy to explain the solution to you in a manner in which you find pleasing. For example 1, let's fall back upon our definition of intersection: we must identify members in BOTH sets.  In this case, the first set actually contains a member -- the empty set itself!  The secondary set, on the other hand, contains no member since it is the empty set.  As such, they've nothing in common.  Thus, their intersection is the empty set.  For example 2, if we follow the definition of a union, a similar conclusion follows -- this time, however, we realize that the union contains members from one set, two sets or BOTH sets.  In this case, the union contains only one element -- The empty set.  Example 3 follows distinctly from the definition of intersection. Let's prove some interesting properties with the empty set then, shall we? Theorem 1. Let A and B be two sets such that B = \( \emptyset \).  Then, A \( \cup \) B = A. Proof.  To prove this, we must show that each side is a subset of the other side.  Thus, we must show (A \( \cup \; \emptyset \)) \( \subseteq \) A and A \( \subseteq \) (A \( \cup \; \emptyset \)). Suppose x \( \in \) A \( \cup \; \emptyset\).  Then, x \( \in \) A or x \( \in \emptyset \).  If x \( \in\) A, then it follows automatically that x \( \in\) A.  If we consider x \( \in\ \emptyset \), then we automatically contradict the definition of the empty set since we're assuming the possibility that there even exists an element in the empty set.  This is absurd, however.  Hence, this is a vacuous truth.  Vacuous truths are almost 'nonsensical' truth statements in a logical context.  Thus, we have shown that A \( \cup \; \emptyset \) \( \subseteq \) A Note: If you study set theory further, the word 'vacuous' will come quite often when referring to empty set.  Feel free to research the word and the context in which it belongs.  Although ProofWiki can be somewhat misleading at times, they have an excellent primer on it here: http://www.proofwiki.org/wiki/Definition:Vacuous_Truth Showing A \( \subseteq \) A \( \cup \; \emptyset \) is quite trivial, and I leave it up to the viewer to try it out.  At best, you will write two sentences; after supposing that x \( \in\) A, you'll find it very easy to showcase that A \( \subseteq \) A \( \cup \; \emptyset \). ,`. Theorem 2. Let A and B be two sets such that B = \( \emptyset\).  Then, A \( \cap \) B = \( \emptyset \). Before we prove this, let us prove a lemma. Lemma 1.1.  Let \( S\) be any set.  Then, \( \emptyset \subseteq \) \( S\). Proof. Suppose this is false.  Then, this implies that there exists an element x \( \in\ \emptyset \) that is not contained in \( S\).  But, this is a contradiction -- the empty set contains no elements.  Hence, the lemma follows. ,`. Now, we can prove theorem 2. Proof. Let x \( \in\) A \( \cap \; \emptyset\).  Then, x \( \in \) A and x \( \in \emptyset\).  This implies that there exists an element which can be found in both sets, but the empty set contains absolutely no element -- hence, how can x \( \in\) in both A and the empty set?  Thus, it is vacuously true that A \( \cap \; \emptyset\) \( \subseteq \emptyset\). The second part of the proof -- \( \emptyset \subseteq \) A \( \cap \; \emptyset\) -- follows from lemma 1.1. ,`. Ah, I wanted to discuss a bit more, but alas ... Feel free to message me with any inquiries regarding this post.  I should hope to come back tomorrow to discuss more and I may even delve into a discussion on the axiom of choice, an extremely controversial axiom for many mathematicians, both amateur and professional alike.

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Second question is, what is the best order to learn mathematics in? I just finished Pre-Calculus, but I feel like the maths taught in school are not entirely satisfying, and I would like to get the complete treatment. Other than that, you have a very interesting blog! I will be looking forward to the next part of your set theory course. Cheers!

I don't know the best order, but I do know I was taught unconventionally.  I was home-schooled and my teaching was a bit more liberal.  For example, I learned Algebra I, II and trigonometry in one entire semester; it was only later during the time I was a sophomore in University that I realized HS split those topics up.  Euclidean Geometry and Precalculus, I learned simultaneously.  This is how I was taught, but not everyone follows that format.  That being said, Mathematics is generally a 'bottom-up' discipline.  To do proofing-based mathematics, it's best to start from set theory and proofing, moving upwards.  It's also suggested that one learn Calculus before moving onto set theory(not because it's a prerequisite, but Calculus makes the brain 'flexible' so to speak).I hope I answered your question(s) in sufficient detail.  Cheers.If you've any more questions, feel free to inquire.

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Greetings! I have a few questions. Well, I'm currently an uprising senior, with lots of spare time. Just recently have I started dabbing into the arts of mathematics. My first questions is, how do I go about understanding texts relating to mathematics? I find myself having some trouble keeping up with a lot of texts. Not sure whether its all of the notation that comes up, or what, but I find this being in the way of me understanding a lot of the things I read. (second question in next ask)

Greetings.In general, when preparing for serious mathematics, it is best to start with a book on set theory and proofing.  In general, most set theory books cover a small portion of logic(conditionals, biconditionals, etc), quantifiers(for all, there exists, negations of those terms, etc), and the 'isms are defined(isomorphism, homeomorphism, etc).  Once these concepts are explicitly defined and comprehended, one can move onto more fulfilling facets of mathematics.  In fact, for my next post on set theory, I plan on talking a bit more about conditionals, quantifiers, logic, etc.  Of course, I strictly talk about proofing-based mathematics; nothing like the stuff you find in basic Calc I, II or III courses where no proofing exists.  For the Calculus series, I merely suggest reviewing some basic Precalculus material(especially trigonometric functions and how they operate) and then moving onto a good book for Calculus.  I learned Calculus from the standard text my University gave us all; the name of the author isn't coming to mind, I'm afraid.  It's considered a standard text at th-- Nevermind, his name is Stewart.  Look up a Calculus text by Stewart.  It's not too rigorous, but it gets the job done.  When you're on a mission to 'relearn' Calculus in a more rigorous fashion, I suggest Spivak's book or Apostal.  Either author gets a thumbs up from me.  Apostal starts with integration, which can perturb some students though.  Fair forewarning.