Looking at the truth table, they're equivalent:  A   B      (A+B)  (A+A'B) ------------------------------- 1   1       1       1 1   0       1       1 0   1       1       1 0   0       0       0   But what manipulation can one do using basic identifies and laws to show that they're the same?
For me, it is pretty clear that $a + \overline{a}*b = a + b$, because the first $a$ in the or will make sure that if the second term must be 'evaluated', $a$ will always be false, and therefore won't matter in the and - probably because I'm a programmer and it is very common to see this unnecessary cluttering in long if-else structures.  I can easily prove it with the truth table as well, just make both expressions', and turns out they are identical.  But I was looking for a 'algebraic' way to prove it. That is, something using the basic operations properties like the fact that they are transitive, commutative, associative, etc. For example, if I was to prove this other simple identity $a + a*b = a$ I would simply do something like:  $a + a*b = a*(1 + b) = a*1 = a$.  Is there a way to also prove the former using this algebraic operations, without resorting to the truth table?
Identity involving Euler's totient function: $\sum \limits_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}$ Let $\varphi(n)$ be Euler's totient function, the number of positive integers less than or equal to $n$ and relatively prime to $n$.    Challenge: Prove  $$\sum_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}.$$   I have two proofs, one of which is partially combinatorial.    I'm posing this problem partly because I think some folks on this site would be interested in working on it and partly because I would like to see a purely combinatorial proof.  (But please post any proofs; I would be interested in noncombinatorial ones, too.  I've learned a lot on this site by reading alternative proofs of results I already know.)  I'll wait a few days to give others a chance to respond before posting my proofs.  EDIT: The two proofs in full are now given among the answers. Logistic regression in R resulted in perfect separation (Hauck-Donner phenomenon). Now what? I'm trying to predict a binary outcome using 50 continuous explanatory variables (the range of most of the variables is $-\infty$ to $\infty$). My data set has almost 24,000 rows. When I run glm in R, I get:  Warning messages:   1: glm.fit: algorithm did not converge   2: glm.fit: fitted probabilities numerically 0 or 1 occurred    I've read the other responses that suggest perfect separation might be occurring, but I'm confident that isn't the case in my data (though quasi-complete separation could exist; how can I test to see if that's the case?). If I remove some variables, the "did not converge" error might go away. But that's not always what happens.  I tried using the same variables in a bayesglm function and got the same errors.  What steps would you take to figure out exactly what's going on here? How do you figure out which variables are causing the problems? How do I take a screenshot on my Mac? What's the easiest way to take a screenshot of my Mac?  I've used Grab.app, but it's a bit annoying just to get a quick screenshot. Is there a faster way? I have a question on Holder space.Is the space $C^2(\mathbb{R}^n)$ embedded in $C^{1,\alpha}(\mathbb{R}^n)$? I think in the bounded domain case, this should be true. But what if in the case where the domain is not bounded? Thanks for any hint! Sequence of sets $f_n$ that converge almost everywhere to $f$ but not almost uniformly to $f$? I can easily find files with given string in name:  me@comp:/usr/local/hydra/hydra-7.4.2$ find -D stat -name "*hack*" ./hack881663129.txt ./hack881663129_7.txt ./hack881663129_5.txt ./hack881663129_4.txt ./hack881663129_4_7.txt ./hack881663129_6.txt ./hack881663129_1_6.txt ./hack881663129_8.txt   How can I also print file details, like size, date of creation etc.? What is the danger of running untrusted software as a normal user Sometimes I have to use some propriety softwares which I don't trust or even pre-compiled open source ones which I don't know if they've been compiled as their source are.   Lets say we installed a propriety application in /opt/software, then we run it as a normal user (without sudo, gksudo, etc).   I know when we run a program it is able to connect to the Internet, monitor X, keystrokes, collect user data and so on. It can't change anything important in the system or infect the system, (because of the Linux permissions system). It is possible for program to add some plug-in/extension to a user's browser (Because browser profiles are available in user home directory). After closing the program and rebooting it can't do anything anymore. Wrong because:   Program can add itself to a shell/file/directory which does the autostart process, so it will run at any login.  If we consider the autostart process clean, without rebooting there is a chance that a process of the program is still active. Programs can delete or modify any file in the user home directory (or any other file which a user has write permission to).   So what should we do?   I think the only safe way to run a propriety program is running it as another user. If running the program as another user somehow is not possible then we should change the autostart files permission so only root can change them. then after closing the program a reboot will do the job.   Am I right? Is there anything that I'm missing? Fatou's lemma and measurable sets I don't know how can I imply Fatou's lemma for any measurable sets $A_k$, that is, $$\lambda(\liminf A_k)\le\liminf\lambda(A_k).$$ How can I prove it? And is there any example in $\mathbb R$ of sequence of measurable sets $A_k$ such that $A_k\subset[0,1]$, $\lim\lambda(A_k)=1$, but $\liminf A_k=\varnothing$ ? Thanks for your help! Showing that $\sqrt[3]{2}\notin\Bbb Q(\alpha_1,...,\alpha_k)$ where $\alpha_i^2\in\Bbb Q\ \forall i$ Let $F$ be the field $\Bbb Q(\alpha_1,...,\alpha_k)$ where  $\alpha_i^2\in\Bbb Q\ \forall i$  I want to show that $\sqrt[3]{2}\notin F$  I thought if $\sqrt[3]{2}\in F$ then $\sqrt[3]{2}\in\Bbb Q\cup(\alpha_1,...,\alpha_k)$ but this is not a disjoint union so I'm not sure what to do... taking the square of $\beta:=\sqrt[3]{2}$ gives $2^{2/3}=\sqrt[3]{4}\notin\Bbb Q$ but that doesn't really advance things  So I'm looking for a hint Is this mold? Black? White? My girlfriend forgot the password for her MacBook (don't ask), so I reset it using the Utilities.  But as you may be aware, this doesn't reset the Keychain password. Her MacBook is now virtually unusable due to Keychain demands continuously provoking a request for the Keychain password.  There are lots of other people asking this around the web and they're usually pointed towards this , which doesn't apply in our situation because the support advice involves entering the old password.  We don't know the old password.  Is it possible to reset Keychain without knowing the old password? We're not bothered about what's in there and what will be lost, we just want the continuous password prompts to go away. I am trying to add smoke to a cigarette. I've created a Cylinder, hit space and chose quick smoke while this sphere was selected. It created a transparent bigger sphere and it is emitting smoke in solid and texture mode. But when I try it out in render mode, it won't show the smoke for some reason. Here is my setup:    The cylinder has a basic white material. Thanks! Determining size of grid cells created by vector grid tool of QGIS? I am working with QGIS 1.8.0 and Im making a vector grid on a map. Does someone know how big are the squares if the grid settings look like in the picture? I dropped my camera while on holiday last summer. Thankfully, my camera wasn't damaged but the kit lens has lost the ability to autofocus and the UV filter and 58mm to 52mm filter step-down (I use it to attach a fisheye filter to the lens) I had on its end are stuck and won't come off.  I've replaced the lens with an 18-200 zoom lens so I don't especially need it. I was wondering if it was worth getting the lens looked at and possibly repaired?  I've heard many places will charge you £50 or so just to look at the lens and given the cost of the kit lens I'm not sure it's worth it. Many companies offer index funds and ETFs. If two funds or ETFs track the same fund, what differentiates the two? Is it common for one index fund to outperform another? How can this happen if the funds/ETFs are tracking the same index?  I found . When forced to choose between them, index ETFs look like the better option (under the right circumstances), but what differentiates index funds/ETFs when compared to similar vehicles? Can I insert an image into an equation? I would like to use a small image in an equation, where a symbol like \alpha might otherwise be. Is there a nice way to do this? Show bigger amplitude of physical pendulum means bigger period Suppose you have a physical . It is true that as amplitude increases, the period increases. Can we demonstrate this fact without explicitly finding the period (which is pretty involved and pretty messy) in:    an intuitive fashion,  rigorously? Show that $\sum_{n=0}^\infty r^n e^{i n \theta} = \frac{1- r\cos(\theta)+i r \sin(\theta)}{1+r^2-2r\cos(\theta)}$ Non-normality in residuals Privilege banner buggy? Or am I just confused? On the main Stack Overflow site, I arrived and had a banner: "Congrats, you've gained the privilege – create wiki posts and 3 other privileges" This confuses me, because I only see two privileges listed at 100%. Is the banner just buggy?  Or is there a better place to see my privileges that the page linked to by that banner?