Consider a set X = $\bigcup_{n=1}^\infty$$S^n$, where $S^n$ = S x S x S x S ... (n times)  Question: Is X countable?   My attempt:   For two countable sets S and T, I found it easy to prove that their Cartesian product is countable. I defined a function f: S x T -> $\Bbb N$,  f(m,n) = $2^m$$3^n$ and proved it was an injection, then defined a function  g: $\Bbb N$ -> S x T, g(n) = (0,n) and showed it was also an injection.   Therefore by Cantor-Bernstein, there exists a bijection. So | S x T | = |$\Bbb N$| and S x T is countable.   Is there some way to extend this idea to the infinite union of Cartesian products of this countable set S ? My first thought was to define some function f($s_{1}$, ..., ) and in some way show that the injection that existed for S x T will always exist for any amount of Cartesian products.  I also know that the for some countable sets A and B, C = B-A --> AUB = AUC and its easy to show that AUB is countable (considering both cases: C is finite or C is infinite and countable)   Completely stuck on marrying the two ideas though! Any help appreciated
While I was reading Enderton's "A mathematical introduction to Logic", I came across the proof of the following sentence: "The set of all finite sequences of members of the countable set A is also countable".   Proof: The set S of all such finite sequences can be characterized by the equation  $$S=\bigcup_{n \in N} A^{n+1}$$ Since A is countable, we have a function f mapping A one-to-one into N. The basic idea is to map S one-to-one into N by assigning to $(a_0,a_1,...,a_m)$ the number $2^{f(a_0)+1}3^{f(a_1)+1}\cdot ... \cdot p_m^{f(a_m)+1}$, where $p_m$ is the $(m+1)$st prime. This suffers from the defect that this assignment might not be well-defined. For conceivably there could be $(a_0,a_1,...,a_m)=(b_0,b_1,...,b_n)$, with $a_i$ and $b_j$ in A but with $m\neq n$. But this is not serious; just assign to each member of S the smallest number obtainable in the above fashion. This gives us a well-defined map; it is easy to see that it is one-to-one.  Note: P is a finite sequence of members of A iff for some positive integer $n$, we have $P=(x_1,...,x_n)$, where each $x_i \in A$.  First of all, I cannot understand why the former assignment might not be well-defined and the latter assignment is well-defined. Secondly, I cannot understand what Enderton means by "just assign to each member of S the smallest number obtainable in the above fashion". By the way, is $(a,b,c,d) = ((a,b),(c,d))$ true? Also, in which cases can I omit/add parentheses in a tuple so as to have an equal tuple?
How to grep for text in a file and display the paragraph that has the text? Below is the text in the file:  Pseudo name=Apple Code=42B state=fault  Pseudo name=Prance Code=43B state=good   I need to grep for "42B" and get the output from the above text like:   Pseudo name=Apple Code=42B state=fault   Does anyone have idea on how to achieve this using grep/awk/sed? 8 AWG solid EGC in raceway for 100 amp subpanel I'm adding a 100 amp subpanel to my detached garage and have a question about the EGC I've already pulled through the underground LFNC-B raceway. Taking the advice from a similar post on this forum, I decided to use a 8 AWG bare solid copper conductor for my EGC in the underground raceway back to the main house. The other 3 conductors are 3 AWG THWN-2 stranded copper. I did double check the NEC tables to ensure these were appropriately sized for the underground feed portion of my run and ensured the conduit fill was less than 40%. What I failed to realize though (until after I won the good battle of pulling this set though the raceway) was that there is a section in the NEC that prohibits the use of solid conductors 8 AWG or larger in raceways: &quot;310.3 Stranded Conductors. Where installed in raceways, conductors of size 8 AWG and larger shall be stranded.&quot; Dangit! Now I'm questioning why I didn't just use insulated green THWN-2 - it would have made my pull much easier! Rookie mistake. So how bad have I goofed up here? Let's say the inspector fails to call this out. Is this a safety issue? What's the rationale for the rule anyways? To prevent numskulls like myself from trying to pull larger gauge solid wire through a raceway and possibly damaging the other conductors? Or is there more to it? Interestingly, there seem to be several exceptions in the NEC that do allow for using 8 AWG solid EGC in raceways (690.46 for Array Equipment Grounding Conductors for example). I'm just trying to get a feel for the severity of this goof-up and if it's worth fixing if not required by the inspector. Any feedback from the experts would be much appreciated! What is the best term to describe the fact that most users have an initial dislike of any change to a UI even though it may really be net better over time?  Note: Question originally appeared How can I vertically center a "div" element for all browsers using CSS? I want to center a div vertically with CSS. I don't want tables or JavaScript, but only pure CSS. I found some solutions, but all of them are missing Internet Explorer 6 support. &lt;body&gt;     &lt;div&gt;Div to be aligned vertically&lt;/div&gt; &lt;/body&gt;  How can I center a div vertically in all major browsers, including Internet Explorer 6? Abount linear functional: If $T(B)$ is bounded, is $T$ bounded? Let $H$ be a infinite dimensional Hilbert space and let $B$ be a basis of $H$ that $H=\overline{span(B)}$  moreover, let $T : H \rightarrow K$ be a linear functional   If $T(B)$ is bounded, is $T$ bounded? The Many Memes of Meta Catchphrases and concepts that spread from person to person are known as [memes], which, courtesy the Internet, can now explode across the Earth like a highly contagious virus (hence &quot;going viral&quot;). As with their real-life counterparts, some infectious diseases are global (pandemic), while others are endemic to specific regions. Stack Overflow, and now even more predominantly Meta Stack Exchange, have seen more than their fair share of these pathological social constructs spread through the user base. They are now ingrained units of our collective culture as SOpedians (a term which I hate, by the way). Just as travelers' immune systems can be assaulted by new diseases in new places, new users are increasingly likely to be miffed by an ingrained meme and left sitting scratching their heads. I therefore propose that this space be used to document the memes endemic to Stack Exchange's culture. Each meme should be listed in a separate answer and I hope that we as a community will be able to provide greater context to each one. Please actually explain each meme in a way understandable to those not already in the know. Especially considering that a lot of these memes aren't really understandable to newer users who weren't around in the early years of the network, or are based on system features that no longer exist.  On other Stack Exchange sites          [memes]: I know cell phone cameras are the lowest rung, but when reading reviews of phones, the reviewers always have sample shots that are much better than what I can manage.  What are some ways to take decent shots with a cell phone camera?  (I'm thinking of the higher end Android or iPhone cameras). Hydrostatic pressure - doesn't density vary with depth? Our class is learning about hydrostatic water pressure and we have been told that we can calculate the force of the liquid on an object at any depth using "the density x 9.8 x the depth". However, as the depth increases, wouldn't the density of the liquid increase because of the weight of the liquid above it compressing it? So should't there be something in the equation to account for the varying density? To me, "density x 9.8 x depth" seems like it is saying that the density will be constant... I tried to rev a file and then pipe it to cat &gt; same_file but it was turning into a blank file.    While I tried rev file.txt | cat &gt; file2.txt &amp;&amp; mv file.txt file2.txt; it worked.    Even rev file.txt | cat &gt;&gt; file.txt; worked.  But when I tried rev file.txt | cat &gt; file.txt it failed. Deleting Individual SE Accounts "Your Sites" in Android left sidebar shown in unpredictable order In the Android app, left sidebar, there is an editable list of "Your Sites". In the editing pane, the sites are re-orderable (although awkwardly), and the order one places the sites in there is preserved, even after quitting the app and returning to this pane. One expects that order to be reflected in the "Your Sites" portion of the left sidebar when the editing pane is closed, but it is not.  I'm not sure what is determining the order of the sites in the left sidebar, it doesn't seem to be by date they were "pinned" into the list one wants to show, either. Show that a positive operator on a complex Hilbert space is self-adjoint Let $(\mathcal{H}, (\cdot, \cdot))$ be a complex Hilbert space, and $A :  \mathcal{H} \to \mathcal{H}$ a positive, bounded operator ($A$ being positive means $(Ax,x) \ge 0$ for all $x \in \mathcal{H}$). Prove that $A$ is self-adjoint. That is, prove that $(Ax,y) = (x, Ay)$ for all $x,y \in \mathcal{H}$. Here's what I have so far. Because $A$ is positive we have $\mathbb{R} \ni (Ax,x) = \overline{(x,Ax)} = (x,Ax)$, all $x \in \mathcal{H}$. Next, I have seen some hints that tell me to apply the polarization identity: $$(x,y) = \frac{1}{4}((\lVert x+y \rVert^2 + \lVert x-y \rVert^2) - i(\lVert x + iy \rVert^2 - \lVert x - iy \rVert^2)),$$ where of course the norm is defined by $\lVert \cdot \rVert^2 = (\cdot, \cdot)$. So my guess is that I need to start with the expressions: $$(Ax,y) = \frac{1}{4}((\lVert Ax+y \rVert^2 + \lVert Ax-y \rVert^2) - i(\lVert Ax + iy \rVert^2 - \lVert Ax - iy \rVert^2)),$$ $$(x,Ay) = \frac{1}{4}((\lVert x+Ay \rVert^2 + \lVert x-Ay \rVert^2) - i(\lVert x + iAy \rVert^2 - \lVert x - iAy \rVert^2)),$$ and somehow show they are equal. But here is where I have gotten stuck. Hints or solutions are greatly appreciated. Adjust space between entries in LoF and LoT What is a NullPointerException, and how do I fix it? What are Null Pointer Exceptions (java.lang.NullPointerException) and what causes them?  What methods/tools can be used to determine the cause so that you stop the exception from causing the program to terminate prematurely? How to do power series expansion What is the coefficient of $x^{11}$ in the power series expansion of $\frac 1{1-x-x^4}$?    How do I do power series expansions? Let $N$ be a normal subgroup of $G$. Let $G/N = \{gN : g \in G \}$. Show that the $(G/N, \circ, 1_{G/N})$ is a group, where:  $\circ : G/N \times G/N \rightarrow G/N$ is defined by $(gN \circ hN) = ghN$ and $1_{G/N} = 1*N$.  Thanks for your help. Proof problem: Show that $n^2-1$ is divisible by $8$, if $n$ is an odd positive integer. Union of subgroups is a subgroup if and only if one subgroup is a subset of the other Let $H$ and $K$ denote two subgroups of a group $G$.  Prove that the union $H \cup K$ is a subgroup of $G$ if and only if $H \subset K$ or $K \subset H$. I have put his on wolfram and obtained answer as follows:  $\sum_{n=1}^{\infty}\frac{n}{2^n} = 2$  And the series is convergent too because $\lim_{n\to\infty} \frac {n}{2^n} = 0$  However I am wondering if there is a convenient way to solve this; I don't think you can represent it by a geometric progression either. So how do we have to do it on paper? Does this recording sound like the pulses emitted by bats? Out of curiosity, I recorded sounds in an area with lots of bats, hoping to be able to &quot;hear&quot; the bats communicating by lowering the ultrasound to a level down to the frequency which can be heard by a human. After a bit of post-processing, I actually hear interesting sounds, but I'm not sure about the origin. It seems that the frequency of the pulses is too low compared to bats. . Are those actual sounds emitted by bats? If not, what other animal could be the source of those sounds? About the audio The sound was recorded at 10 p.m. at  near a river, with more of a dozen bats flying in this zone. There is a bridge nearby, and the sides of the river are flat concrete, which means that there are a lot of opportunity for sound reflection (although high frequency sounds don't reflect well). Zoom H5 with the default XY directional microphone was used. The bitrate was set to 24-bit. Unfortunately, I used the sampling frequency of 48 kHz instead of the maximum 96 kHz. The following transformations were made to the original audio:  The pitch was shifted by -12 semi-tones, which means that the 20 kHz sound becomes 10 kHz, and 15 kHz becomes 7.5 kHz.   All sounds below 6 kHz were removed.  The sound was amplified.  Severe noise reduction was applied to eliminate everything but the actual sounds emitted by the bats.   The empty parts of the audio were removed.  The sound was amplified once again.   Frequency The frequency analysis of the transformed audio a peak around 9 kHz (18 kHz in the original audio), slowly fading to suddenly drop at 16 kHz (32 kHz in the original audio).   contains many sources with different indications about the frequency range of the sounds emitted by bats, only a few referencing the frequencies low enough to be heard by a human:  Frequency Produced 10kHz-120kHz Ramsey, Gabriel, McGuirk, Phillips, Watenpaugh. Holt Physical Science. New York: Holt, Rinehart and Winston Publishers, 1986.  while most agreeing on the fact that bats use frequencies higher than 20 kHz.  Most bats use frequencies in the range 20-80 kHz, only a few bats use frequencies less than 20 kHz. Long, M.R. &amp; Savage, J.G. Mammal Evolution. New York: Harper, 1976: 108.  The observed range of 18..32 kHz seems too low compared to the sources, but I have an impression that the frequencies vary a lot between different species of bats. Pattern When viewing the audio track (see below), each cry consists of several pulses in two to three sub-groups which are in turn in two groups. Each peak is a bit less than 1 ms. The second group occurs from 40 to 200 ms. after the end of the first one. Each sub-group is separated by 5 to 10 ms. The frequency of every pulse doesn't vary, except the very first pulse. The first cry starts by a more quiet pulse at 6 kHz (12 kHz in the original audio). The second and third cries don't follow this pattern.  indicates that there are three categories of sounds emitted by the bats:  Short clicks,  Frequency-swept pulses,  Constant frequency pulses.   It looks like it's still the third category which we are observing here, and the overall pattern looks similar to the one described for the bats. The gap between the left and the right track is mostly irrelevant: at higher frequencies, microphones become more and more directional, and the sound is reflected less, which leads to such difference.     Note: this question was considered to be on-topic at Biology.SE on .