There exists a bijection between $(0,1)$, $(0,1]$ and $[0,1]$?  These 3 sets are not countable and since there are all in $\mathbb{R}$ they should have the same number of elements, so my question which are those bijective functions?  I've been thinking, but I cannot find of any.
How to define a bijection between $(0,1)$ and $(0,1]$?   Or any other open and closed intervals?   If the intervals are both open like $(-1,2)\text{ and }(-5,4)$ I do a cheap trick (don't know if that's how you're supposed to do it): I make a function $f : (-1, 2)\rightarrow (-5, 4)$ of the form $f(x)=mx+b$ by \begin{align*} -5 = f(-1) &amp;= m(-1)+b \\ 4 = f(2) &amp;= m(2) + b \end{align*} Solving for $m$ and $b$ I find $m=3\text{ and }b=-2$ so then $f(x)=3x-2.$  Then I show that $f$ is a bijection by showing that it is injective and surjective.
Where [if anywhere] should the apostrophe go in "three days work"? Which is correct?     1 Three day's work      2 Three days' work      3 Three days work   I would probably guess (2) is right, since the work belongs to the three days ("three days of work"). But I'm not sure. Would it be different if it was only one day? Glassware with scratched/etched surfaces generate more bubbles (, ). Why these rough surfaces allow easier heterogeneous nucleation? If space of bounded operators L(V,W) is Banach, V nonzero, then W is Banach (note direction of implication) Let $V,W$ be normed vector spaces, and $L(V,W)$ be the space of bounded linear operators. Usually I would only see the statement "If $W$ is Banach, then $L(V,W)$ is Banach.". But  writes that there is a converse: "If $L(V,W)$ is Banach, and if $V$ is non-trivial, then $W$ is Banach". This is pretty interesting since I never seen a converse before. I was wondering if anyone has a nice proof.  I tried reversing the proof for the usual direction, but the inequalities can't be reversed. I also tried to start with a Cauchy sequence in $W$, and construct linear operators (using Hahn-Banach to control the operator norms), but alas I cannot say much about the distance between these operators (much less say that it is Cauchy) String Comparison in Java As per the following example, and as in  In bash, where has the trailing newline char gone?, I want to know "why" it happens    x="$(echo -ne "a\nb\n")" ; echo -n "$x" | xxd -p  # Output is: 610a62  # The trailing newline from the 'echo' command #   has been "deleted" by Command Substitution   I assume there must be some very significant reason for a shell action, namely Command Substitution, to actually delete some data from the command output it is substituting... but I can't get my head around this one, as it seems to be the antithesis of what it is supposed to do.. ie. to pass the output of a command back into the script process... Holding back one character seems weird to me, but I suppose there is a sensible reason for it...  I'm keen to find out what that reason is... Here $n$ belongs to natural numbers. Firstly, I proved the relation by putting $n = 1$ . Then, taking $$|\sin(mx)| \leq m|\sin x|$$ true, I had to prove $$|\sin(m + 1)x| \leq (m + 1)|\sin x|$$  Now, here I got stuck. How to prove it?? Please help. What is the best way to convert seconds into (Hour:Minutes:Seconds:Milliseconds) time? What exactly is a bad backlink that is Disavow worthy? I have been reading Webmasters SE and many other blogs on this topic but I could not understand how do we truly evaluate a bad link that we must disavow in GWT. Everybody seem to have a different opinion on this.  A little background  I am a WordPress developer and 3 years ago I gave away a WordPress theme to my blog visitors which has a dofollow link back to my business website. Like this.  WordPress theme designed by &lt;a href="http://domain.com"&gt;MyBusinessName&lt;/a&gt;   From last 2 years I was not paying any attention on organic traffic to my business website because of other projects.  That free WordPress theme helped us getting ~10K backlinks from variety of domains and websites. All most 99% backlinks are from non-related websites and in many different languages.  Now  Last week it came to my attention that in October 2013, organic traffic from Google started decreasing and in just 2 months it became as low as 25% of October. Before October it was gradually increasing ~5-10% per month.  So I researched and draw to the conclusion that it could be because of Penguin Update. Since  was introduced at the same time.  Now I need information on what a bad/toxic link is.  I was about to disavow almost all links but after reading many topics on Webmasters SE I think I need more information on how to identify a bad link.   Does credit link in footer consider as spam because it build thousands of backlinks? Do foreign language website backlinks consider as spam? Should I remove all backlinks from theme users because almost all are from non-related content domains? Although some links from non-related content domains have nice DA/PA. Should I remove those too? I understand foreign language anchor text backlink could be a spam but what if it is from related foreign language content.   The question is how exactly one can find a link that is bad and must be removed. What are beginner’s books on typography? $x^3+x+1$ irreducible over $\mathbb{F}_2$, $\theta$ root, compute powers of $\theta$ Problem statement:     Show that $x^3+x+1$ is irreducible over $\mathbb{F}_2$ and let $\theta$ be a root. Compute the powers of $\theta$ in $\mathbb{F}_2$.   I am having trouble computing the powers of $\theta$. $x^3+x+1$ is irreducible, since any cubic polynomial over a field is reducible if and only if it has a root in the field. Since $\mathbb{F}_2=\{0,1\}$, we see that none of these is a root of $x^3+x+1$.  To compute the powers of $\theta$, note that $\theta^3+\theta+1=0$. Hence $\theta^3=-(\theta+1)$. Is this a good enough answer for $\theta^3$?  To compute $\theta^2$, note that $\theta = -(\theta^3+1)$. Therefore, $$\theta^2=(\theta^3+1)^2 = \theta^6 + 2\theta^3 + 1 = (-\theta-1)^2-2\theta-2+1,$$ which just reduces to $\theta^2$. Any ideas for computing these powers?  Thanks! I just installed a new version of Ubuntu and I want to roll it back to the previous version.  How can I do that? Is that even possible? I'm trying to install Wifite from , but I can't install cowpatty.  See below:  root@vitor-hp:~# apt-get install cowpatty Reading package lists ... ready Building dependency tree Reading state information ... ready E: Could not find package cowpatty root@vitor-hp:~#    Translated from Portuguese  I use Ubuntu 13.10 64-bit. 2018 monthly product team updates Mid-2017  started publishing monthly updates. This post will continue that effort into 2018. (.)  Updates are posted at the beginning of the month. They cover what was done last month, what is still in progress and new work that will be started in the current month.  Updates:      (CoC, nav, OpenID, responsive design, bugs, campaigns)  (CoC, comment flags, nav, OpenID)  (question wizard, nav, user survey, OpenID)  (tag watching, GDPR)  (responsive design, tags, teams)  (responsive design, OpenID, teams, sponsorship)  (responsive design, facebook auth, question lists, teams)  (newsletter, sign up, Documentation reputation, channels)  (top bar issues, hats, question template) InDesign: How do I automatically align the bottoms of different columns of text boxes to the same height as the one with the most text? In InDesign, I have some filled text boxes, sized by columns, with the one on the left being three columns wide.   The wide text box is most likely to contain the most text (it's an invoice template).  I will use the 'Fit frame to content' button on the Selection Tool toolbar for automatically vertically resize it to fit the containing text.  How do I make the other three smaller text frames automatically adjust to the same height as the big (and longer) text box? This is my circuit. I need to find the power dissipated by Vlamp. So the lamp turns on if the Vlamp is more than 2V. When the lamp is on it has an internal resistance of 10 Ohms, and when it is off it acts like an open circuit.  How would I find the power dissipated? I know how to do it for Voltage sources and Resistors, but not elements like these.  One more thing, I already figured out that: Battery has open circuit voltage of 4V R1 = 5 Ohms R2 = 5 Ohms k = 0.2A/V When lamp on and Vlamp = 2V, Rbat = 10 Ohms   . Is infinite regress of causation possible? Is infinite regress of causation necessary? For a number of reasons &mdash; including perhaps a desire to feel that we have a complete understanding of where we came from, or at least an understanding which is completely sufficient for all of our purposes &mdash; there is a strong tendency to suppose that an infinite regress of causes and effects is impossible. We see this for instance in the . Traditionally in Greco-Roman influenced philosophy, this prime mover has been identified with the notion of a creator god. A more modern formulation comes from the Big Bang theory, which (although we study the question) happened for no reason that we know of; and perhaps occurred for no reason at all. Indeed, if time is a property of the universe, a good argument can be made that .  At the same time, we have gotten a lot of leverage in science from the ideas that everything happens for a reason, i.e.&nbsp;because it is caused by something else. This is sometimes described as the . Aristotle also applies this idea, except that the first mover is made an exception, something which exists literally for no reason. Indeed, the emotionally negative way that "no reason" is used in common speech, as something groundless and irrational, indicates how widespread the feeling is that the idea of uncaused events is deeply dissatisfying.  Almost all of our modes of critical thinking are infused with both of these ideas: that we may work from first principles, a definite starting point, counting up from zero (or one, historically); but also that we may trace the causes of things to some point, and then later ask how that starting point came to be. But the two ideas are themselves in conflict.  Which is true &mdash; that infinite causal chains are impossible? Or that they are necessary? Or are they perhaps possible without being necessary?   Related questions: How can I disable my internet connection from terminal? BOOLEAN FUNCTIONS: generate a function $f: \{0, 1\}^n \rightarrow \{0, 1\}$ from a $n \times n$ S-Box I want to calculate the hamming weight of a S-Box using this formula: $\text{hw}(f) = \sum_{x=0}^{2^n-1} f(x)$. Where $f: \{0, 1\}^n \rightarrow \{0, 1\}$  My problem is that I don't know how to get the $f$-function.  I found that helpful paper: . At Page 9 (18) boolean functions are described. But she only prints the truth table (Table 2.1) without describing her $f$-function. She only says $f$ is a linear function, therefore I think $f$ depends on my S-Box?  For example, if I have this $2 \times 2$ S-Box:  0    1    2    3  1    3    0    2    $00 \rightarrow 01\\ 01  \rightarrow 11\\ 10  \rightarrow 00\\ 11  \rightarrow 10$  What are the $f$-functions and what is the hamming weight? They agreed to stay connected for hours in their provisional support group, looking for answers in their counterparts.   I want to change the bold part to say that they all shared the same problem.  Might end up using one of the above, but I feel like I need some more options to weigh in.    Edit for context: they all share the same rare cognitive impairment Here's a question I've been asked;  Given distinct primes $p,q,r$, show that any group $G$ of order $pqr$ is not simple.   So far, my idea has been to individually check each possible proper subgroup, since by Lagrange there are only $$\dbinom{3}{2} + \dbinom{3}{1} + 1$$ of them, and show that at least one of them other than the trivial group is normal. Surely there is a smarter way, since generalizing this approach wouldn't work so well if I wanted to show similar results for groups who's order is a product of $n$ distinct primes.  Is there some other approach? Thanks for any insights