What is the proof that total number of subsets of a set is 2^n?

I have known "by heart" that total number of the subsets of a set of n number is \(2^n\) I was struggling to find an intuitive explanation, and two answers helped me to understand it.

For each element, you have two choices: either you put it in your subset, or you don't; and these choices are all independent.

Citation:

Bruno Joyal (https://math.stackexchange.com/users/12507/bruno-joyal), What is the proof that the total number of subsets of a set is $2^n$?, URL (version: 2013-10-31): https://math.stackexchange.com/q/546417

Another was this table that represented the sets as binary, to denote 0 as an absence of the element and 1 as presence of the element.

https://dl.dropbox.com/s/0smcrv98o8igrhh/power_set.jpg

That can help us understand that the total number of subsets of set of n is \(2^n\)

2018 in Books

In 2018, I read 19 books that I have kept track via GoodReads. The app shares fancy page that gives interesting details about my reading in 2018. Here is my Year In Books page.

These were some notable books that I had read.

  • Lauren Ipsum - A Story about Computer Science and Other Improbable Things by Carlos Bueno
  • Apache Mesos Essentials by Darmesh Kakadia
  • Beyond the Door by Philip K Dick
  • Manga Guide to Linear Algebra by Shin Takahashi
  • Guns, Germs and Steel by Jared Diamond
  • Chaos: Making of a new science by James Gleick
  • Yummy Sourashtra Recipe Book by my parents
  • Astrophysics for people in a hurry by Neil deGrasse Tyson
  • The Mysterious affairs at Styles by Agatha Christie
  • The Hackers Diet by John Walker
  • Code: The Hidden Language of Computer Hardware and Software by Charles Petzold
  • The Best we could do by Thi Bui
  • I am Home, the potraits of immigrant teenagers by Erica McConnell, Rachel Neumann and Thi Bui
  • The Myth of a strong leader: Political leadership in the modern age by Archie Brown
  • Stranger in a strange land by Robert Heinlein

I have organized the reviews of these under the blog tag books-read-in-2018.

Book Review: Stranger In a Strange Land

Stranger in a Strange LandStranger in a Strange Land by Robert A. Heinlein
My rating: 5 of 5 stars

Wow, what a book. It goes into multiple topics like being human, politics, religion, love, sex, community, spirituality, entrepreneurship, money, influence, negotiation, strategy and "understanding" of it all.

This gave a new word "grok" to English speaking people, which roughly means to "understand thoroughly" and becoming one with the concept.

One interesting thing about this science fiction is, it combined the real world concepts of business, stocks, politics, and ownership with imaginary situation where humans have already traveled to mars and there human of raised by martian among us. The details covering the rise of Mike to the status of a godman, and formation of his communion was fun read.

Just imagine how close is this quote "Thou art god, I am god. All that groks is god.” by Valentine Michael Smith to Sanskrit Verse "Tat Tvam Asi".

The character Jubal Harshaw, attributes that one can explain away everything by holding on solipsism, and pantheism.

I was impressed with Robert Heinlein covering so many topics. I really enjoyed it.

Book Review: The Myth of the Strong Leader - Political Leadership in the Modern Age

The Myth of the Strong Leader: Political Leadership in the Modern AgeThe Myth of the Strong Leader: Political Leadership in the Modern Age by Archie Brown
My rating: 5 of 5 stars

This book covers a great deal on political leadership and helps the reader to understand the political landscape of multiple nations. The crux of this, it is not a strong leader that brings a net positive change, but a leader who is flexible, who understands the situation well, goes into the details, establishes that the processes, and follows it through that will bring the net positive change. The way, the different leaders have accomplished is quite varied, and depends on the situation and the time.

This book introduces some amazing real life characters like, Adolfo Suárez of spain who transformed the country to Democracy from the previously established monarchy in a peaceful manner. It also shares how Deng Xioping is the person behind the economic advancement of China from 1970 onward. It shares the perspective on how Fidel Castro overthrew the dictatorship in his country, and how lived by his principles, and did not accumulate any wealth for himself.

One thing that struck with me, when this book was written, Obama was the president of USA, and even then, the system in place in USA was such that, no one particular person was ultimately responsible everything in the government. It has been understood that answer for "who controls the whitehouse of USA", seems " no one knows". This is not a snarky comment, but a statement that workings of the government in USA is very complex, and there are multiple forces in effect at any time, including political parties, supporters, lobbyists, and corporations with interests, and it will be too much to give credit and put focus on a single person.


This book covers lot. It covers Britain, USA, Europe, South America, South Africa, China, Russia, touches upon New-Zealand, India and other nations. The author explores how various leaders in these countries have shaped the political landscapes of these places for better.

Book Review: I am Home portraits of immigrant teenagers

I am Home portraits of immigrant teenagersI am Home portraits of immigrant teenagers by Erica McConnell, Rachel Neumann and Thi Bui
My rating: 5 of 5 stars

Memorable book. It covers the stories of teenagers various different parts of the world like Keren, Eritrea, San Salvador, Huehuetenango, and lots of places around the world, who are now studying in Oakland International High School, and share what they think, when they think of "Home".
Book does a very good job of portraying the diversity that exists in OIHS.

My Model

I was thinking about this model while I was introspecting. I decided to capture it, so that I can share this idea.

https://dl.dropbox.com/s/xkajrkz93qymip2/Screenshot%202018-12-09%2007.57.59.png?dl=0

Book Review: The Best We Could Do

The Best We Could DoThe Best We Could Do by Thi Bui
My rating: 5 of 5 stars

This is a personal memoir of the author, Thi Bui, as she tries to recollect her own story of coming to the US as a refugee from Vietnam on a boat.

Fast forward many years, the author is well educated, accomplished in life, settled in society, and has a family of her own. When she has her first child, she tries to associate herself with her mother and her times in Vietnam and their family migration to the US.

It covers the political history of Vietnam well, and how the ordinary people suffered, without any exception, both at the hands of external oppressors and from local oppressors alike. There was no way out, except to Run away. And, run they did, in search for stabler societies so they and their children at-least can have a chance to live.

It is a heart-wrenching story as she understands the story of her own mother, father, and the reason they become the way they are, the hardships and life that they have seen in other parts of the world, the risks they took, and ultimately, the best they could do for their children.

It is a family story, and it helps us understand our own parents better.


Quotes that I liked from the book

“This - not any particular piece of Vietnamese culture - is my inheritance: the inexplicable need and extraordinary ability to run when the shit hits the fan. My refugee reflex.”

― Thi Bui, The Best We Could Do

...

“To understand how my father became the way he was, I had to learn what happened to him as a little boy. It took a long time to learn the right questions to ask.”

― Thi Bui, The Best We Could Do

Book Review - Code: The Hidden Language of Computer Hardware and Software

Code: The Hidden Language of Computer Hardware and SoftwareCode: The Hidden Language of Computer Hardware and Software by Charles Petzold
My rating: 5 of 5 stars

This is a highly influential book in the subject of how Computers work from ground up. They author deals with the subject and introduces computers as if they were a natural evolution of inventions that had already taken effect and had proved useful to the society.

After introducing the various needs for communication, the author explains how it can achieved using bulbs, and then chips, and then how it is stored and done at the higher scale using Computers. He goes both in the Hardware as well as Software section of how computers work, and why we need operating systems and programming languages.

The section on float point arithmetic caught my attention and imagination completely. This was a lucid explanation on storing decimal valued numbers using binary digits.

I throughly enjoyed this, and is already helping me appreciate some of the basic concepts used in Computers and Computer Science.

Book Review: The Hacker's Diet

The Hacker's Diet: How to Lose Weight and Hair Through Stress and Poor NutritionThe Hacker's Diet: How to Lose Weight and Hair Through Stress and Poor Nutrition by John Walker
My rating: 5 of 5 stars

This is the first health book that I could not put down. This is the book that is helping me to manage my weight and my health. So, it is a life-saving book for me.

I was worried about my weight gain, and my lack of time to do exercises. I have 2 kids now and have an hour commute to my work. Both of these leave me with very less time to do exercises regularly, and I was constantly feeling worried about weight gain.

This book helped to break the problem into two.

a) Managing Weight.
b) Doing Exercises.

For a), I completely understood how this can be controlled by paying attention to the Calorie intake. That' it. I adjusted my calories and I went back to -3KG within 3 weeks. I have goal to go -13 kgs further so that I can maintain my weight at a desirable number. I am into managing of my weight now.

Once that is done, I know the need and utility value for exercises and I will able to make time for it to live for longer so that I could continue doing whatever I like.

Rarely, self-help books have proved valuable to me. This the 2nd self-help book that I can say that, it has proved valuable to me and the concepts taught will remain with me for the rest of my life.

Thank you, John Walker.


Lisp - Two Bit History Article

Excellent article on History and Concept of the programming language Lisp was published at this site called twobithistory

https://twobithistory.org/2018/10/14/lisp.html

Author goes in into the details on History of Lisp, the motivation behind the design, how the language, and an early book "SICP" created a culture of veneration for that language.

He also credits Paul Graham, and XKCD comics to have significantly surged the interest back in lisp.

Interesting anecdotes like how braces end up being chosen for the language. The motivation of the SICP book was very noble.