boolean ring .commutative ring
How to show that every Boolean ring is commutative?
How can I concatenate two arrays in Java? How can I read inputs as numbers? Lets say we have a dependent variable $Y$ with few categories and set of independent variables.    What are the advantages of multinomial logistic regression over set of binary logistic regressions (i.e. )? By set of binary logistic regression I mean that for each category $y_{i} \in Y$ we build separate binary logistic regression model with target=1 when $Y=y_{i}$ and 0 otherwise. How can I intercept XMLHttpRequests from a Greasemonkey script? Proof of $\gcd(a,b)=ax+by\ $ [Bezout's identity] The Cartesian product of a finite number of countable sets is countable The Cartesian product of a family $(A_i\mid i\in I)$ is defined as $$\prod\limits_{i\in I}A_i=\{f:I\to\bigcup A_i\mid f(i)\in A_i\}$$ Let $I_n=\{i \in \mathbb N \mid i \le n\}$ for $n\in \Bbb N$ and $A_i$ be countable for all $i\in I_n$. Then $\prod\limits_{i\in I_n}A_i$ is countable.   My attempt: Lemma 1: $(\prod\limits_{i\in I_n}A_i) \times A_{n+1}$ and $\prod\limits_{i\in I_{n+1}}A_i$ are equinumerous for all $n \in \mathbb N$. (I presented a proof ) Lemma 2: If $A$ and $B$ are countable, then $A \times B$ is countable. (I presented a proof ) I will prove the theorem by induction on $n$. The theorem is trivially true for $n=0$. Assume that the theorem is true for $n=k$, then $\prod\limits_{i\in I_k}A_i$ is countable. Since $\prod\limits_{i\in I_k}A_i$ is countable (by inductive hypothesis) and $A_{k+1}$ is countable, then by Lemma 2 $(\prod\limits_{i\in I_n}A_i) \times A_{n+1}$ is countable. Furthermore, by Lemma 1 $(\prod\limits_{i\in I_k}A_i) \times A_{k+1}$ and $\prod\limits_{i\in I_{k+1}}A_i$ are equinumerous, then $\prod\limits_{i\in I_{k+1}}A_i$ is countable too. Thus the theorem is true for $n=k+1$. This completes the proof.   Does this proof look fine or contain gaps? Do you have suggestions? Many thanks for your dedicated help! Will spoiled food always make you sick? Create a page for useful links? Modern electronics industry has developed very efficient LEDs with nice and fancy colors (“true green” , “cyan”, and “blue”) based in InGaN technology, or similar. Some of these LEDs are specified as having forward voltages of 3.2 to 4.2 V. Here are examples of such LEDs, , or surface-mount .  But the usual MCUs are powered from 3.3 V supply, and the GPIO output is usually at 3.1-3.2 V, which seems to be barely enough to drive these LEDs . How can I use these LEDs as indicators of some signal statuses, or power rail status? Do I need a series resistor R1 here?    If yes, how do I choose its value?  DISCLAIMER: This "question-answer" comes as elaboration of the original question , to address the particular use case of LEDs as logic state indicators. Do Wine Viruses only work while Wine is running? I heard Linux can get viruses through Wine,and I was curious if the viruses work only while Wine is running.  In other words, could I stop a virus from doing it's thing by just quitting Wine? Difference in sentence meaning between present continuous and simple present constructions Family name pluralization When pluralizing family (last) names that also happen to be common English words, does the pluralization follow the same rules as the common word?  For example, "the Smith family" can be pluralized as "the Smiths", but what if the family name is "Wolf" or "Fish"?  Would the correct pluralization be "the Wolfs" or "the Wolves"? give space above and below a DIV Catching poisoned arrows When catching an arrow tipped with poison (like the ones used by yuan-ti) with an ability like the one the monk has or the one given by arrow catching gloves: does the ability reduce the poison damage? If so, is the total damage or the piercing one caused by the arrow only the damage that must be downed to 0 to catch the missile? How to extend USB beyond the recommended maximum cable length of 5m? I'm planning to use USB port for an application(from a device to a PC) but the distance between the device and the PC will be around 15m. The recommended maximum USB cable length is 5m.  How can I buffer the data without loosing any information, safely? Would a USB hub work? Computing $\lim_{n\rightarrow \infty} \frac{1+\sqrt{2}+\sqrt[3]{3}+\cdots+\sqrt[n]{n}}{n}$. I have a problem with the calculation of the following limit. \begin{equation} \lim_{n\rightarrow \infty} \frac{1+\sqrt{2}+\sqrt[3]{3}+\cdots+\sqrt[n]{n}}{n} \end{equation} I do not know where to start! Thank you very much Can I see things that I've said in chat that have been starred? I was reviewing the star on chat.SE list this morning and I noticed that quite a few of the items on the list were mine.  I was curious as to how often people enjoy the things I post there enough to star them, but I can't seem to find statistics about this anywhere.  Mana told me that no such feature exists.  (the best we could come up with is ) So here's my request - can I filter/search the chat history for things I've said that other people have starred? When I'm generating a List of Figures in my project, all figures are grouped by chapter. I would like to have the whole list without any additional new lines between different chapters.  How it looks like:  Figure 1.1  Figure 1.2  Figure 2.1  Figure 3.1  Figure 3.2   How it should:  Figure 1.1  Figure 1.2 Figure 2.1 Figure 3.1  Figure 3.2   I tried to apply the following solutions but failed completely :        My main file:  \documentclass[12pt,oneside,titlepage,utf8x]{report}  \usepackage{times,a4wide,multirow,tabularx,graphicx}  \usepackage{psfrag,fancyhdr,longtable}  \usepackage[T1]{fontenc} \usepackage{float} \usepackage[utf8]{inputenc}  \usepackage{url} \usepackage{color} \usepackage{verbatim} \usepackage{fancyvrb} \usepackage{natbib} \usepackage{relsize}  \usepackage{caption} \captionsetup[longtable]{belowskip=10pt}  \usepackage[left=4cm,top=2cm,right=3cm,bottom=2.5cm]{geometry}  \renewcommand{\chaptermark}[1]{\markboth{\nouppercase{\thechapter.\ #1}}{}} \pagestyle{fancy} \headheight 30pt \rhead{} \lfoot{} \cfoot{\thepage} \rfoot{}  \setlength{\parindent}{2em} \setlength{\parskip}{1em} \linespread{1.3}   \begin{document}  \begin{titlepage} \maketitle \end{titlepage}  \tableofcontents  \chapter{First} \label{sec:First} \input{chapter1/chapter1}  \chapter{Second} \label{sec:Second} \input{chapter2/chapter2}  \chapter{Third} \label{sec:Third} \input{chapter3/chapter3}  \relsize{-1} \listoffigures \normalsize  \bibliography{praca_mgr} \bibliographystyle{plain} \nocite{*}  \end{document} Computers on board Starfleet vessels are quite primitive compared to on board computers such as . At their best, they can decipher requests such as "Computer, delete that last sentence" and utter things such as "That crewman is no longer on board this vessel". Even real life computers will be capable of this within a decade or so (by some estimates). Even computers such as HAL 9000 are likely to be available within a few decades.  So, in a genre where intelligent computers are abound, why this strange absence? Did the Federation run into some Battlestar-Galactica-like problem with AI in the past and decide to keep on board computers decentralized?  But if that is the case, how can one explain Data and highly intelligent holographic characters? The technology and programming constructs clearly exist, yet Starfleet vessels continue to be manned by hundreds of people standing in front of consoles. How to know that your machine learning problem is hopeless?