Given cardinal numbers $A$ and $B$. Does there hold  $$A&gt;B \quad\Rightarrow\quad 2^A&gt;2^B\quad?$$  In other words: is the cardinality of a set uniquely determined by the cardinality of its power set? My first guess was Yes. But then I thought about the continuum hypothesis and that without it there might be a cadinal $\aleph_1$ with $\aleph_0&lt;\aleph_1&lt;\mathfrak c$. I was not sure if $2^{\aleph_1}$ falls nicely between $\mathfrak c$ and $2^{\mathfrak c}$. So it might depend on CH? I have not much experience with cardinal arithmetic, so no clue how to think about this.
Is there any example that for cardinal numbers $\kappa &lt; \lambda$, we have $2^\kappa = 2^\lambda$?  My guess is that it only depends on whether GCH holds. Is it true?
Example sentence:     I love when your dog just lets you sit there to pet them. You don’t necessarily know if they are enjoying it, but they love you enough to just sit there with you for a bit.   Is this correct? We assume the words "you" and "your" refer to the speaker of the sentence, and not to the listener, as second-person usually does. But it also refers to dog owners in general. I have always been curious about this. From official :     Due to a head injury, Cyclops is unable to shut off his optic blasts   at will and must therefore wear ruby quartz lenses to block the beams.   From the same page:     Cyclops possesses the mutant ability to project a beam of heatless   ruby-colored concussive force from his eyes, which act as   inter-dimensional apertures between this universe and another.   When his eyes are always blocked by ruby-colored beam of concussive force, how does he see? Book series my dad told me about. It involves space travel, clones, and an indigenous people with kangaroo tails My dad read a book series that he recommended to me last year. Now that I am ready to start it, neither of us can recall the title. He said there were several in the series but he can't recall how many.   The book is about traveling to a new planet, presumably with the goal of colonizing it. The crew are clones who have no memory. I think the goal was for the older original versions of the clones to transfer their consciousness into the clone bodies to be able to live on the new planet but something goes wrong.   There is an indigenous species on the planet who are humanoid but have kangaroo like tails.   The clone crew has markings on them that they don't understand until late. Apparently they are markings to indicate status or role on the new world. The person who they voted as there leader, it turns out, is marked as a servant or slave class. How to modify the system default background image I am about to ship a new 10.04 installed system to a customer. This is the 1st system that we are shipping with our preinstalled software and an additional hardware test platform, so I thought it would be a nice idea to have the desktop wallpaper include our company's logo.  What I want to do is just to add the logo to the default background image. the questions are:   Where are the set of backgrounds stored? What are the rules for creating a custom background from scratch? If I create a background from scratch, how do I add it to the system's background set (i.e., not just save it in the user's Pictures directory - I want it to be available to all future users of the machine)? What is the best word describing the nature of expiring in a short amount of time, connoting "losing public attention"? Replacing all values in a table > $T$ How do I transfer programs from my old computer to my new Windows 7 computer? I know about the Easy Transfer tool included in Windows 7 that will transfer all my files but I want to transfer my programs. Also, I don't want to buy any gimmicky program if I don't have to. It's my understanding that "easily" is the regular adverb for "easy", e.g. "He makes friends easily".  But in some collocations one still uses "easy", e.g. "Take it easy", "Rest easy" or "Slow down easy". Is there any rule when "easy" would be used over "easily", or does one just have to know these collocations? My windows directory is huge I have always had problems with my windows directory's growing to a huge size.  It started with windows 95 then 98 then 2000 and xp.  Windows 7 however is absolutely huge.  The windows folder itself with out program files is 21.4 gigs...add the program files and other windows files and it is a whopping 32 gigs.  My question is why is it so large and what can I do about it.  At the rate this is going I will be out of space in now time. What do you call the functions that make a composite function.  Example : $e^{\sin x} $ is made up of $\sin x$ in the argument of $e^x$  So what are $\sin x$ and $e^x$ called here in this context? Basic functions or something?   (Please don't tell me the former is trig and latter is exponential function, I know that, but that's not what I'm asking here, please try to understand)  Hope I made myself clear. Apply the latest Service Pack and Cumulative Update Don't heavier objects actually fall faster because they exert their own gravity? The common understanding is that, setting air resistance aside, all objects dropped to Earth fall at the same rate. This is often demonstrated through the thought experiment of cutting a large object in half. The halves clearly can't then fall more slowly just by being sliced in two.  However, I believe the answer is that when two objects fall together, attached or not, they do "fall" faster than an object of less mass alone does. This is because not only does the Earth accelerate the objects toward itself but the objects also accelerate the Earth toward themselves. Considering the formula: $$ F_{\text{g}} = \frac{G m_1 m_2}{d^2} $$  Given $F = ma$ thus $a = F/m$, we note that the mass of the small object doesn't seem to matter as when calculating acceleration the force is divided by the $m$ term, its mass. However, this overlooks that the force is actually applied to both objects, not just to the smaller one. The acceleration on the second, larger object is found by dividing $F$, in turn, by the larger object's mass. The two objects' acceleration vectors are exactly opposite, so closing acceleration is the sum of the two:  $$ a_{\text{closing}} = \frac{F}{m_1} + \frac{F}{m_2} $$  Since the Earth is extremely massive compared to everyday objects, the acceleration imparted on the object by the Earth will radically dominate the equation. As the Earth is $\sim 5.972 \times {10}^{24} \, \mathrm{kg} ,$ a falling object of $5.972 \, \mathrm{kg}$ (just over 13 pounds) would accelerate the Earth about $\frac{1}{{10}^{24}}$ as much, which is one part in a trillion trillion.  Of course in everyday situations, we can for all practical purposes treat objects as falling at the same rate because of this negligible difference—which our instruments probably couldn't even detect. But I'm hoping not for a discussion of practicality or what's measurable or observable, but what we think is actually happening.  Am I right or wrong?  What really clinched this for me was considering dropping a small Moon-massed object close to the Earth and a small Earth-massed object close to the Moon. This made me realize that falling isn't one object moving toward some fixed frame of reference, but that the Earth is just another object, and thus "falling" consists of multiple objects mutually attracting in space. Let $x_n$ be a sequence of the type described above. It is not monotonic in general, so boundedness won't help. So, it seems as if I should show it's Cauchy. A wrong way to do this would be as follows (I'm on a mobile device, so I can't type absolute values. Bear with me.) $$\left|x_{n+1} - x_n\right| \leq \frac{1}{n^2}.$$ So, we have $$ \left|x_m -x_n\right| \leq \sum_{k=n}^{m} \frac{1}{k^2}$$ which is itself Cauchy, etc., etc. But, of course, I can't just use absolute values like that. One thing I have shown is that $x_n$ is bounded. Inductively, one may show $$ \limsup_{n \to \infty} x_n \leq x_k + \sum_{k=n}^{\infty} \frac{1}{k^2},$$ although I'm not sure this helps or matters at all. Thanks in advance.  Disclaimer I've noticed that asking a large number of questions in quick succession on this site is often frowned upon, especially when little or no effort has been given by the asker. However, I am preparing for a large test in a few days and will be sifting through dozens of problems. Therefore, I may post a couple a day. I will only do so when I have made some initial, meaningful progress. Thanks. The following question is from Artin's Algebra.      If $A$ and $B$ are two square matrices with real entries, show that $AB-BA=I$ has no solutions.    I have no idea on how to tackle this question. I tried block multiplication, but it didn't appear to work. How to correctly install Minecraft, including setting any needed packages or apps to install it and to troubleshoot the most common problems that Ubuntu users face when trying to use Minecraft.  I am trying to install and run Minecraft without first having to install the Oracle version of Java. I want to use the OpenJDK version that comes with Ubuntu and as a last resort, use the Oracle one.  I also want to know a list of problems related to Minecraft in Ubuntu and the possible solutions to them.  A step by step guide on how to install Minecraft would also help. So, I'm trying to figure out how to run the scene from the beginning of Lost Mine of Phandelver (5e starter set) where the party is ambushed by goblins and I'm really confused as to how Passive Wisdom (Perception) works.  The included rulebook mentions on page 6 :     When you hide, there’s a chance someone will notice you even if they aren’t searching. To determine whether such a creature notices you, the DM compares your Dexterity (Stealth) check with that creature’s passive Wisdom (Perception) score, which equals 10 + the creature’s Wisdom modifier, as well as any other bonuses or penalties. If the creature has advantage, add 5. For disadvantage, subtract 5. For example, if a 1st-level character (with a proficiency bonus of +2) has a Wisdom of 15 (a +2 modifier) and proficiency in Perception, he or she has a passive Wisdom (Perception) of 14.   So the party is being ambushed by goblins and at that point the DM guide mentions on page 7 :     Check to see who, if anyone, is surprised. [...] Make a Dexterity (Stealth) check for the goblins, rolling once for all of them. Roll a d20, add the goblins' Stealth skill modifier (+6) to the roll and compare the result to the characters' Passive Wisdom (Perception) scores.   So to me this reads like I should roll 1d20+6 for the goblins and compare that to the static Passive Wisdom (Perception) score on the character sheet of a PC. The PCs don't get to roll anything as they're not actively looking for enemies, even though they are investigating the dead horses.  I would also think that if the characters were supposed to roll, the DM guide would clearly instruct so, given that this is the starter set.  However, watching  at around 8:01 I notice that the DM is explicitly asking his party for a perception roll. And the same thing happens on a couple of other videos where people are running the Lost Mine adventure. The DM always requests a roll.  So what gives? Shall I take it that the DMs ask for the roll because they consider the party to be actively seeking for hidden foes? Or am I just misreading the rulebook and the DM guide? I have been thinking of getting a flash to improve my low light photography (particularly indoors). However, there seem to be a number of different models available at similar price points (budget of INR 8,000-12,000 ~$150-250). What features and functionality should I look for to choose a suitable flash?  I'm using a Canon 550D with the 18-55mm IS &amp; 55-250mm IS lenses. Metz seems to be one of the brands easily available in India (beside the OEM flashes). Currently, I am considering the Metz 36 AF-5, 44 AF-1 and 50 AF-1, which seem to be considerably cheaper than their Canon counterparts. Complex number question on proving an inequality. If $|z_1|=1,|z_2|=1$, how can one prove $|1+z_1|+|1+z_2|+|1+z_1z_2|\ge2$ Importing a Single Symbol From a Different Font How can I prevent Overflow for the Gamma function? I have a Dirichlet distribution from which I'm maximizing $\alpha$ by calculating the log likelihood with the following equation  $p\left(s|\alpha\right) = \sum\limits_{i=1}^O\log\left(\frac{n_i!}{\prod\limits_{j=1}^kn_{ij}!}\frac{\Gamma\left(\alpha_0\right)}{\Gamma\left(\sum\limits_{j=1}^k\left(n_{ij}+a_j\right)\right)} \prod\limits_{j=1}^k \frac{\Gamma\left(n_{ij}+\alpha_j\right)}{\Gamma\left(\alpha_j\right)}\right)$  I have a python script to calculate the log likelihood. It works for small values, but when $n_{ij}$ gets high $\Gamma\left(\sum\limits_{j=1}^k\left(n_{ij}+a_j\right)\right)$ and $\Gamma\left(n_{ij}+\alpha_j\right)$ python runs into overflow error (the max for math.gamma is between 171 and 172).   Is there a way to change the likelihood function so that $\Gamma\left(\sum\limits_{j=1}^k\left(n_{ij}+a_j\right)\right)$ and $\Gamma\left(n_{ij}+\alpha_j\right)$ don't run into overflow errors. Can I just divide the values in $\Gamma$ by for example 100, if I then compare the likelihood between $p(s|\alpha)$ with different $\alpha$ values do the best $\alpha$ values still have the highest likelihood?