coinomega

Being a monopoly (or to be precise: to be in a dominant position), is not problematic in EU competition law. It is the abuse of said position that is illegal!

https://ec.europa.eu/competition/publications/factsheets/ant...

Very simple script, and only confirms what has been posted several times here.

However interesting scenario when the game master reveals a door with a goat, and the player does not know if the game master consciously decided to remove that particular door. In this scenario, probability of winning swings from 0.33 to 0.67.

Aware subscript k is not displayed correctly on mobile, but formatting on Medium is generally not great when it comes to formulas

> "Bitwise" is really shorthand for "this is an operation that acts on the representation of a number in base-2, rather than on the number itself"

I would argue that while bitwise operations are indeed more easily represented/computed in base 2 than in base 10, they do act on the 'number itself', whichever base you then represent this number in. ... at least in the approach I took, where I treat bit strings as representations of 'numbers', specifically positive integers. If you think of such operations as manipulating 'strings' of characters - whatever character set those are taken from - rather than numbers then indeed we indeed arrive to different conclusions, and your digit-wise MIN generalization makes sense.

I think both approaches are meaningful, although conceptually different, and I have tried, however clumsily, to explain the 2 approaches in a previous post (https://medium.com/biffures/bits-101-120f75aeb75a#.fc93no6od) before settling for the 'number' approach for this post.

I am actually not a proponent of using base 10 as the standard view for bitwise operations, I merely use that base to show that the function looks complicated in base 10 and brings little further intuition on what pattern AND follows. Note that the rest of the article and all visualizations are NOT base-dependent and do not use the base-10 formula. (I did write my numbers in base-10 in the sketches simply out of convenience, but feel free to translate to any base).

Is this revolutionary? Certainly not. But I do think the graphs look cool, and I was happy to share them. Hope this clarifies some of the thinking behind the post; and thanks for the constructive thoughts! - seems like I can do plenty of read-up .-)

C

∧ is also the notation for bitwise AND

http://csrc.nist.gov/publications/fips/fips180-4/fips-180-4....

I used the notations found in FIPS 180-4 http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.180-4.pdf

Assuming you actually care about the answer, this just help me drew the exploratory graph, and this property was useful to fill in cells for which operands were equal. Finding obvious properties is a great way to learn a pattern without even computing things. Unlike what you may think, I had no idea what I was about to find - I had never seen this chart drawn before, and so this post is structured around how I thought about the exploration, not around how it may be presented to an audience who already knows way more than I do on bitwise operations. Sounds fair?

The function which to any integer a associates f(a, a) with f defined as in the article is the identity function on the natural integer set.

corrected

I don't think you need to be so unforgiving?

If you want extra precision, a -> f(a, a) is Id.

Knowing that helped me understand one property of the chart, never said that was the most stunning of properties, and the best articulated one :-)

Fair, changed paragraph to "While the result is easily computed and the process easily understood in base 2 (i.e., using the ‘bit strings as language’ lens), the AND function written in base-10 notation looks in fact non-trivial:"

The point of the article is to switch bases (exploring the many facets).

Per my previous thoughts, "the binary notation is a contingency, useful to understand and define specific functions that will run fast on computers given their binary architectures. However, because the notation does not convey meaning in itself, those numbers and functions are as well described in any arbitrary base, namely base 10 for humans, or base 16 (hexadecimal) for conciseness."

Hello Chris, author here!

AND is not multiplication in base 2 :)

The point of the article was to make sense of the apparent complicatedness of the math formula through nice visuals that help humans understand what AND does number-wise.

I may have used the terms math vs. numbers loosely, but this seemed to be the terms most people would understand - and a vocabulary used in other places (Wikipedia, though maybe hardly a reference?). I have tried to explain my "bit string" vs "number" approach [in this other article](https://medium.com/biffures/bits-101-120f75aeb75a#.gz4ka3t7k) if you are interested.

Really not implying that boolean algebra is not math though - and I am pretty sure this has nothing to do with the 'poor state of math education'.