Prove that the set of made of the following formulas is feasible (Materials on specific logic problems) I'm looking for some free resources online (books, courses, text, sites and so on) on how to solve these two particullar problems and understand the material behind them. There isn't any material on this in my native language and it seems quite hard to find any info when the I don't know the exact terminology in english.   Any help with these problems will be helpful :)    1)  Prove that the set of  made of the following formulas is satisfiable ( realizable / I don't know the exact terminology in english)  $ \forall x \neg p(x,x)$  $ \forall x \forall y ( p(x,y) \Rightarrow \neg p(y,x))$  $\exists x \forall y (x \neq y \Rightarrow p(x,y))$  $\exists x \exists y ( x \neq y \Rightarrow p(x,y))$    2)  The structure $A$ has a bearer $N$ (the set of natural numbers) and it is for a language with only one non-logical symbol $p$, which is a predicate interpreted like : $&lt;n,k,m&gt; \in p^A \leftrightarrow  n^2 = km + 1$ . Prove that $\{0\}$, $\{ 1 \}$ and $\{ &lt;n,n&gt; : n \in N \}$ are definable.    Agian any help on finding material on how to solve these and understand them is of great help. Thanks :)
What are the prerequisites for studying mathematical logic? I am looking to study mathematical logic, however, I find that introductory books are very daunting, which kind of disheartens me. You see, slowly but surely, I started to realize that the maths which I have learned did not just pop out of thin air, but is a collection of systems, which must of been developed via some other system, i.e, maths did not develop itself.  So I began to look into the origins of mathematics, and read that it was developed via a type of logic, which exists sort of by 'default', via a set of axioms, and then of course I looked up the definition of axioms.   So given that I'd be studying a type of logic whose origins are self evident axioms, naturally I believed there would be no prerequisites. However, in looking up mathematical logic, I have come across things such as Boolean algebra, sets, first order logic, some other type of logic, called 'traditional logic', as well as references to a sort of calculus, though not in a mathematical sense, I think.  So all in all, I am trying to develop a type of mental spider web, and I am trying to find out the strands which lye at the absolute bounds so that I may learn this mystique logic. Though I have no idea where to start.  Side note: This is the book I have started reading:   Credit goes to Wolfgang Rautenberg.
Automatic image format detection in PHP I am looking for a way to take a user uploaded image that is currently put in a temporary location ex: /tmp/jkhjkh78 and create a php image from it, autodetecting the format.  Is there a more clever way to do this than a bunch of try/catching with imagefromjpeg, imagefrompng, etc? Should Emergency Funds be Used for Infrequent, but Likely, Expenses? Say we break down non-monthly costs into the following three categories.   High level of predictability: This would include oil changes that may be needed every 2 to 4 months depending on usage, car insurance (if paid every 6 month), car registration, etc. Medium level of predictability: There are some expense which one can reasonable assume will occur on an infrequent basis.  For example, car repairs, house repairs, or even doctor visits during flu season. Low level of predictability: For example, losing a job, significant medical expense, etc.   Which of these categories are emergency funds meant to cover?  I think it is pretty clear that a high level of predictability costs should not use emergency funds while a low level of predictability should.  But medium level of predictability costs seem different, and it left me wondering what is generally defined as an "emergency".  While you do not know when these costs will occur or how much they will be, you could devise some pretty good rules of thumb on frequency and cost. Say you know your 2007 Volvo should have a repair cost averaging out to $100 per month.  Conceptually you could save $100 each month and when your car breaks, not have to spend any emergency funds on the repair.  Is doing this "double-saving" and not suggested if you already have the 6 month emergency fund saved up?  Note: this is a question on how to save money for expenses.  It is not a question about if you should spend emergency fund money to fix a car when you have no other money set aside. Git command to show which specific files are ignored by .gitignore I am getting my feet wet with Git and have the following issue:  My project source tree:  / | +--src/ +----refs/ +----... | +--vendor/ +----...   I have code (currently MEF) in my vendor branch that I will compile there and then move the references into /src/refs which is where the project picks them up from.  My issue is that I have my .gitignore set to ignore *.dll and *.pdb.  I can do a git add -f bar.dll to force the addition of the ignored file which is ok, the problem is I can not figure out to list what files exist that are ignored.  I want to list the ignored files to make sure that I don't forget to add them.  I have read the man page on git ls-files and can not make it work. It seems to me that git ls-files --exclude-standard -i should do what I want.  What am I missing? How can I retrieve \thepage (or detect a page change) from Lua in LuaTeX? I am writing (contributing to) a LuaLaTeX package where a Lua function is used to generate images that are inserted in the LaTeX document.   One piece of information we need is the current page number to determine if we're on an odd or even page (I know that this is not that trivial, and I know about some strategies beyond \thepage). We set a Lua variable when we're still in the LaTeX part, and so far this works pretty well.  However, now it turns out that the sequence of images we create may be distributed on several pages, and we would really like to reflect that because our images may behave differently on odd and even pages.  Basically what we have is a series of  tex.print('\\includegraphics{something}\\par')   statements, executed in a loop within one Lua function.  Now the question: is there any way to either directly detect a page change between two such graphics or repeatedly retrieve the current page within such a loop?  All the solutions I have found for determining current odd or even state are in LaTeX itself. I know how to use \directlua to retrieve a value for use in LaTeX, but is there a way to do the opposite - retrieve a LaTeX value from within LUa?  Edit: to respond to TeXnician's comment and answer: When I modify the example code so it produces repetitions of paragraphs it seems that the page counter won't get updated within one Lua chunk:  \documentclass{article}  \newcommand\printluapage{%     \directlua{for i = 1, 60 do tex.print(tex.count['c@page'])          tex.print([[\par]]) end} }  \begin{document} \printluapage  New command  \printluapage  New command  \printluapage \end{document}   The page number is only updated with each new call of \printluapage. Intersections of all subgroups is a nontrivial subgroup, so every element has finite order. I need help to prove this result:  "Let $G$ be a group such that the intersection of all its subgroups other than $\{1\}$ is a subgroup different from $\{1\}$. Then all its elements have a finite order".  I know I must think of an element $g$ of infinite order. That will imply that every subgroup has infinite order (because it will contain an element like $g^k$). After this, I don't know which step I can take. Can someone help? The reCAPTCHA has become too difficult for humans to solve The reCAPTCHAs have suddenly become almost illegible. I regularly have to change it four or five times before I find one I can read. I am tempted to just enter the known word (an easy guess), and leave the other. Can something be done? what does ∇ (upside down triangle) symbol mean in this problem Given $f(x) = \frac{1}{2}x^TAx + b^Tx + \alpha $   where A is an nxn symmetric matrix, b is an n-dimensional vector, and alpha a scalar. Show that  $\bigtriangledown _{x}f(x) = Ax + b$   and  $H = \bigtriangledown ^{2}_{x}f(x) = A$   Is this simply a matter of taking a derivative with respect to X, how would you attack this one? $x^3+x+1$ 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! Quarks in a hadron: where does the mass come from? How can I know what speed card to get for my camera? Cameras rarely specify exactly what speed cards you should be using in them - I just checked the manual for my DSLR, and all it says is to use an SDHD card, and a "high speed" card should be used for recording video.  On the other hand, there are a number of different rating systems for memory card speed (33x! 150x! Class 2! Class 10!) and price seems to increase with speed.  Is there any way to test the maximum write speed the camera is capable of?  Are the read/write speeds documented anywhere? Disproving the statement "If $f(z)$ is not an entire function, then $g(z) = f^2(z)$ is not entire.” If $f(z)$ is not an entire function, then $g(z) = (f(z))^2$ cannot be an entire function.”  From this statement why would saying &quot;Let $f(z) = \sqrt z$, assuming that it is a branch that takes $−1$ to $i$. Then it is not analytic on the branch cut, but $g(z) = (f(z))^2 = z$ is obviously an entire function.&quot; Not be a proper example to disprove this statement/ be a incorrect counterexample? I understand that for $\sqrt z$ to be a incorrect counterexample means it is not analytic on the branch -1 to I, therefore not entire. However I thought $\sqrt z$ would be analytic throughout the whole branch cut from -1 to i. Since on the complex plane, $\lim_{z \to -1} \sqrt z$ exists as it would approach $i$ in that case, and in the other case $\lim_{z \to i} \sqrt z$ it approaches $\sqrt i$ meaning that at least on that branch cut in the complex plane it is continuous everywhere, therefore the partial derivatives exist everywhere on the branch cut meaning it is analytic, but this assumption seems to be incorrect. What am I missing or not understanding here. Precisely, I don't understand why $\sqrt z$ would not be a proper counterexample to the statement. Uninstall kubuntu-desktop I installed kubuntu-desktop using sudo apt-get install kubuntu-desktop but I no longer want to use KDE.  How do I remove all the packages that were installed by this command ?  Thank you Intersect (analysis) of polygons generating empty output Confusing [...] List in Python: What is it? So I was writing up a simple binary tree in Python and came across [...]  I don't believe this to be related to the Ellipsis object, more it seems to have something to do with an infinity loop (due to Python's shallow copy?). The source of this infinity loop and why it doesn't get expanded while expanding when accessed is something I'm completely lost to, however   &gt;&gt;&gt; a [[[[[], [], 8, 3], [[], [], 3, 2], 6, 3], [], 1, 4], [[], [], -4, 2], 0, 0] &gt;&gt;&gt; Keys(a)#With a+b [0, 1, 6, 8, 3, -4] &gt;&gt;&gt; Keys(a)#With [a,b] [8, [...], [...], 3, [...], [...], 6, [...], [...], 1, [...], [...], -4, [...], [...], 0, [...], [...]] &gt;&gt;&gt; Keys(a)[1]#?? [8, [...], [...], 3, [...], [...], 6, [...], [...], 1, [...], [...], -4, [...], [...], 0, [...], [...], 8, [...], [...], 3, [...], [...], 6, [...], [...], 1, [...], [...], -4, [...], [...], 0, [...], [...]]     Version using a+b  def Keys(x,y=[]):     if len(x):y+=[x[2]]+Keys(x[0],y)+Keys(x[1],y)#Though it seems I was using y=y[:]+, this actually outputs an ugly mess     return y   version using [a,b]  def Keys(x,y=[]):     if len(x):y+=[x[2],Keys(x[0],y),Keys(x[1],y)]     return y   So what exactly is [...]? What are Virtual Methods? Why would you declare a method as "virtual".   What is the benefit in using virtual? What is Reflection in Java Can anyone explain? How many Shutter Clicks on a New Camera considered normal? How many Shutter Clicks on a New Camera considered normal?    I purchased a quotes new Nikon D750 and checked the shutter count and it has 264 Shutter Clik count on it.    Is that suspect of a returned purchase or used camera? LaTeX for Microsoft Surface I have my new Microsoft Surface and I was wondering whether it's possible to install pieces of software like MikTeX and TexMaker on this tablet. Does anyone know? This question was asked in an entrance test for an undergraduate mathematics program today, held all over India.   Question: $f$ is a differentiable function in $[0,1]$ such that $f(f(x))=x$ and $f(0)=1$.  Find the value of $\int_{0}^{1} (x-f(x))^{2016} dx$.  I tried to solve it by substituting $f(f(x))$ in place of $x$, but could not proceed much further. Any hints or solutions will be appreciated. I am trying to convince my friend that the integral of $0$ is $C$, where $C$ is an arbitrary constant. He can't seem to grasp this concept. Can you guys help me out here? He keeps saying it is $0$.