How can I prove $\Bbb N$x$\Bbb N$ is countably infinite? Is the proof in some book about sets? Somebody help please.
How can I prove that the Cartesian product of two countable sets is also countable?
Change default shell for Terminator How to remove all files in a directory except hidden ones? Set of continuity points of a real function I have a question about subsets $$ A \subseteq \mathbb R $$ for which there exists a function $$f : \mathbb R \to \mathbb R$$ such that the set of continuity points of $f$ is $A$. Can I characterize this kind of sets? In a topological,measurable or in some way? For example, does there exist a function continuous on $\mathbb Q$ and discontinuous on the irrationals? Is this noun used as an adjective? I read this recently in :     At the end of the summit, the French and European officials had   claimed a points victory over the Germans by getting them to agree   more firmly to a target date of January 1st next year to entrust the   European Central Bank (ECB) with the ultimate authority to supervise   the euro zone’s 6,000-odd banks.   "Points" is a plural, and I suppose it doesn't function as a noun in this sentence since it's following the article "a", so is it an adjective adorning "victory"? Getting from Denver Amtrak to Fort Collins The eastbound Amtrak arrives Denver 6:38 PM (18:38).  How can I get to Fort Collins from there?   The  runs  from Denver International Airport to Fort Collins, but the airport is 40 km from the Amtrak station, so that's a major detour, that would mean taking bus FX from Market Street Station 7:18 PM, arriving at the airport at 8:27 PM, then the super shuttle 9:05 PM, arriving Fort Collins CSU at 10:55 PM; almost 4 hours after departing from Denver. Greyhound Lines appears to have a bus departing 12:45 AM, arriving 02:00 AM — although probably as quick as it gets, this is not exactly a comfortable time. RTD Denver has transportation to Longmont and  from Longmont to Fort Collins, but  appears to depart 7:05 PM — so I won't make it there when arriving by Amtrak.  A taxi would cost an estimated .   Is there any other way to get to Fort Collins when arriving by the eastbound Amtrak, any way that I've missed? Is it possible to link multiple leaders lines to one piece of annotation?  I'm working in ArcGIS 10.2 Desktop Advanced with non-feature linked geodatabase annotation. I have hundreds of numbers labeling various polygons, but some polygons share a number. Instead of creating multiple numbers/annotations for each polygon I'd like to have one number with multiple lines pointing to each polygon.   So, in the image below, the number 5 corresponds to multiple polygons. Here I've moved 5 from it's ideal location to show that the second leader line is not connected to the number annotation. My workaround has been to create another piece of geodatabase annotation with a blank text string to serve as a placeholder for a leader line. The other option was to use map/graphic annotation to create the second leader line, but with the multiple disadvantages of not being in the same annotation feature class as the number annotation.    My goal would be for each piece of number annotation to be connected to multiple leader lines like appears below. Question of an isomorphism of $\epsilon_ 0$ and a subset of the rationals. $1^n +2^n + \cdots +(p-1)^n \mod p =$? Calculate for every positive integer $n$ and for every prime $p$ the expression  $$1^n +2^n + \cdots +(p-1)^n \mod p$$    I need your help for this. I don't know what to do, but I'll show you what I know.   Wilson's theorem The identity $X^p-X = \prod_{a \in \mathbb{F}_p-1}(X-a)$ $\exists a \in \mathbb{Z}, \ a^2 \equiv -1 \mod p \qquad \iff \qquad p \equiv 1 \mod 4$   Now can you please provide me a hint? Is this a dismissal by hit wicket? Consider the following situation,     A batsman's bat breaks while hitting the ball after which a piece of the   bat flies in the air, hits the stumps and the bails fall down.   Is this a dismissal by hit wicket? What is the proper use of [square brackets] in quotes? I sometimes see square brackets used while quoting.  My assumption is that they are replacing a pronoun with what the object of the pronoun, but I never know for sure because I don't usually get to see what the original quote looks like before the modification.  What are these called and what are the rules of use? Suppose we want to define a first-order language to do set theory (so we can formalize mathematics). One such construction can be found . What makes me uneasy about this definition is that words such as "set", "countable", "function", and "number" are used in somewhat non-trivial manners. For instance, behind the word "countable" rests an immense amount of mathematical knowledge: one needs the notion of a bijection, which requires functions and sets. One also needs the set of natural numbers (or something with equal cardinality), in order to say that countable sets have a bijection with the set of natural numbers.  Also, in set theory one uses the relation of belonging "$\in$". But relation seems to require the notion an ordered pair, which requires sets, whose properties are described using belonging...  I found the following in Kevin Klement's,  (pages 2-3).  "You have to use logic to study logic. There’s no getting away from it. However, I’m not going to bother stating all the logical rules that are valid in the metalanguage, since I’d need to do that in the metametalanguage, and that would just get me started on an infinite regress. The rule of thumb is: if it’s OK in the object language, it’s OK in the metalanguage too."  So it seems that, if one proves a fact about the object language, then one can also use it in the metalanguage. In the case of set theory, one may not start out knowing what sets really are, but after one proves some fact about them (e.g., that there are uncountable sets) then one implicitly "adds" this fact also to the metalanguage.  This seems like cheating: one is using the object language to conduct proofs regarding the metalanguage, when it should strictly be the other way round.  To give an example of avoiding circularity, consider the definition of the integers. We can define a binary relation $R\subseteq(\mathbf{N}\times\mathbf{N})\times(\mathbf{N}\times\mathbf{N})$, where for any $a,b,c,d\in\mathbf{N}$, $((a,b),(c,d))\in R$ iff $a+d=b+c$, and then defining $\mathbf{Z}:= \{[(a,b)]:a,b\in\mathbf{N}\}$, where $[a,b]=\{x\in \mathbf{N}\times\mathbf{N}: xR(a,b)\}$, as in  question or  on Wikipedia. In this definition if set theory and natural numbers are assumed, then there is no circularity because one did not depend on the notion of "subtraction" in defining the integers.  So my question is:     Question Is the definition of first-order logic circular?   If not, please explain why.   If the definitions are circular, is there an alternative definition which avoids the circularity?   Some thoughts:   Perhaps there is the distinction between what sets are (anything that obeys the axioms) and how sets are expressed (using a formal language). In other words, the notion of a set may not be circular, but to talk of sets using a formal language requires the notion of a set in a metalanguage. In foundational mathematics there also seems to be the idea of first defining something, and then coming back with better machinery to analyse that thing. For instance, one can define the natural numbers using the Peano axioms, then later come back to say that all structures satisfying the axioms are isomorphic. (I don't know any algebra, but that seems right.) Maybe sets, functions, etc., are too basic? Is it possible to avoid these terms when defining a formal language? jquery selector not working with [ ] How can one prove the statement $$\lim_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution.  This is homework. In my math class, we are about to prove that $\sin$ is continuous. We found out, that proving the above statement is enough for proving the continuity of $\sin$, but I can't find out how. Any help is appreciated. In the proof of the lemma     For every limit ordinal $\alpha$, $\textrm{cf } \alpha$ is a regular cardinal    Jech says that     It is easy to see that if $\alpha$ is not a cardinal, then using a mapping of $|\alpha|$ onto $\alpha$ one can construct a cofinal sequence in $\alpha$ of length $\leq |\alpha|$, and therefore $\textrm{cf } \alpha &lt; \alpha$.    I've spent a while trying to construct this cofinal sequence and was wondering if I was on the right path:   From a bijection $f$ between $\kappa = |\alpha|$ and $\alpha$ take $S = \{\beta \in \kappa : \ f(\gamma) &lt; f(\beta)$ for all $\gamma &lt; \beta\}$ then if $\xi$ is the order type of $S$ and $g$ is the isomorphism from $\xi$ to $S$ the function $f\circ g$ is a cofinal sequence in $\alpha$ and $\xi &lt; \alpha$.   Is this on the right track, or is there an easier way of doing it? How long does it usually take for published articles to show up on Google Scholar? The title pretty much says it all - what's the typical delay with which Google Scholar indexes newly published articles? (Note I'm not talking about citations, but about the article showing up at all.)  Edit: the article in question went for over six months after being published (in a well-regarded Springer journal with its own Wikipedia page) without appearing on Google Scholar, until it received a citation, at which point it was added to Google Scholar almost immediately. One more motivation to write papers that get cited. :)  Edit 2: Another article in an even more mainstream journal took a little over seven weeks. 'innerText' works in IE, but not in Firefox Multiple chrome.exe processes in Task Manager I know that each new tab on Google Chrome have its own process, but I have 14 chrome.exe processes running actually as I can see it on the Task Manager, and only two tabs are opened on Google Chrome.    Could it be due to Chrome extensions? What other reason can open more chrome.exe process and how to avoid it? undefined control sequence when using \addchap When I compile my thesis.tex file which is compiled fine including the \addchap, I noticed that I have used \usepackage{unnumberedtotoc}. When I want to use my thesis.tex file on another computer it says tex capacity exceeded, sorry [input stack size=5000]. When I uncomment \usepackage{unnumberedtotoc}, it says ! undefined control sequence  \addchap.  Here I have pasted all the beginning of the packages that I have used in my thesis but I still get some problems in compilation, I don't know why, but I think if I can compile this code, then I can solve the problem  :   \immediate\write18{makeindex \jobname.nlo -s nomencl.ist -o \jobname.nls} \documentclass[12pt]{report} \DeclareMathSizes{12}{30}{16}{12} \usepackage[utf8]{inputenc} \usepackage{graphicx} \usepackage{lineno} \usepackage{amsmath}  \usepackage{array, nccmath}  \usepackage[nottoc]{tocbibind} \newcolumntype{L}{&gt;{$}l&lt;{$}} \newcolumntype{m}{&gt;{\displaystyle}l} \usepackage{framed} \usepackage{cclicenses} \usepackage{makecell} \usepackage[french]{babel} \usepackage{setspace} \usepackage{mathptmx}  \usepackage{multirow}  \usepackage{gensymb}  \usepackage{empheq}   \usepackage{mathtools}   \usepackage{array}   \renewcommand{\arraystretch}{1.9}   \newcolumntype{P}{&gt;{\centering\arraybackslash$}p{1.6em}&lt;{$}}   \newcolumntype{C}{&gt;{\centering\arraybackslash$}c&lt;{$}}   \usepackage{makecell}   \usepackage{booktabs}   \usepackage{subcaption}   \usepackage[intoc]{nomencl}   \makenomenclature   \setcounter{secnumdepth}{4}   \setcounter{tocdepth}{4}   \usepackage[justification=centering]{caption}    \usepackage{etoolbox}     \usepackage{unnumberedtotoc} %https://github.com/johannesbottcher/unnumberedtotoc/     \usepackage[automark,     plainheadsepline,     headsepline,     plainfootsepline,      footsepline,     markcase=ignoreupper]{scrlayer-scrpage}    \clearpairofpagestyles    \ihead*{\leftmark}    \cfoot*{\pagemark}    \setkomafont{pageheadfoot}{\scshape}    \usepackage{titlesec}    \graphicspath{{imagesthesis/}}    \def\frenchlistfigurename{Liste des figures}    \usepackage[a4paper,right=20mm,left=25mm,top=26mm, bottom=26mm]{geometry}    \makeatletter    \titleformat{\chapter}[display]    {\normalfont\huge\bfseries}{\chaptertitlename\ \thechapter}{20pt}{\Huge}    % this alters "before" spacing (the second length argument) to 0    \titlespacing*{\chapter}{0pt}{0pt}{20pt}    %\patchcmd{\@makechapterhead}{\vspace*{20\p@}}{}{}{}% Removes space above \chapter head    %\patchcmd{\@makeschapterhead}{\vspace*{20\p@}}{}{}{}% Removes space above \chapter* head    \makeatother    \title {itle of the thesis}    \titleformat{\paragraph}    {\normalfont\normalsize\bfseries}{\theparagraph}{1em}{}    \titlespacing*{\paragraph}     {0pt}{3.25ex plus 1ex minus .2ex}{1.5ex plus .2ex}     \usepackage[thin, , thinc]{esdiff}  \usepackage{subcaption}  \usepackage{caption}  \usepackage{framed}    \usepackage{nomencl}   \setlength{\nomitemsep}{-\parskip}\usepackage{booktabs,multirow}  \renewcommand{\cellset}{\renewcommand{\arraystretch}{0.5}}   \begin{document}  \renewcommand\arraystretch{0.8} %\noindent  \doublespacing  \maketitle  \tableofcontents %\listoffigures  \addchap{Publications et communications}   \section*{$\:$  a. Publications:}  \section*{$\:$  b. Communications:}  \addchap{Remerciements}  \end{document} Is it okay to record audio for a lecture? Sometimes the Professor will say things quickly, or say things before I can write them down. It would be of great benefit if I could record audio for the course, but I am not sure the legal guidelines around doing so. MISSISSIPPI problem How many arrangement of the letters in MISSISSIPPI have at least 2 adjacent S's?  I was thinking that I can glue two of the S's together, so there will be 9 letters plus the special letter SS, and the number of arrangements will be $10!/(2!2!4!)$, this is not the correct answer, but what is wrong with my reasoning.  (the correct answer is $11!/(4!4!2!)-7!/(4!2!)\binom{8}{4}$, I understand the solution)