Every field is integral. Is this proof valid?  Suppose R is not integral →exists a∈R where a is a zero divisor →exists x∈R, x not 0 such that a.x=0 But R is a field so a is invertible →exists a'∈R such that a.a'=1 Now that a.x=0 and a.a'=1 a(a'-x)=1 →a'-x is the inverse of a →a'-x=a' →x=0 contradition
The proof I have starts of with $\;xy=0\;$ in a field.   Then $x^{-1}$ exists because it is a field.  Then $x^{-1} xy=x^{-1} 0$.  Therefore $y=0$.  But surely if an integral domain can not have any zero divisors, how can we end the proof by saying $y=0$? Surely then y is a zero divisor and hence the field is not an integral domain?
Proof of "the continuous image of a connected set is connected" None of the existing questions is exactly answering my question so I'm posting a new question, but feel free to refer me to some already answered question!   In Rudin Theorem 4.22, we know that      If $f$ is a continuous mapping of a metric space $X$ into a metric space $Y$, and $E$ is a connected subset of $X$, then $f(E)$ is connected.    In the proof, we started with consider $f(E) = A \cup B$, where $A$ and $B$ are nonempty separated subsets. Then put $G = E \cap f^{-1}(A)$ and $H = E \cap f^{-1}(B)$. Then Rudin is claiming that $E = G \cup H$. I'm a little suspicious about this. What if $f$ is non-surjective, then $f^{-1}(A) \cup f^{-1}(B)$  is only a proper subset of $E$? Is there a property of $f$ being continuous that forces $f$ to be 1-1? Calculate $E(X^{2n})$ where $X$ is normal (0,1) I need help proving the following:  Let $X$ be normally distributed with parameters $\sigma=0$ and $\mu=1$. Let $n$ be a positive integer. Show that:  $$E(X^{2n})=\frac{(2n)!}{2^nn!}=:(2n-1)!!$$  I've tried the change of variables $Y=X^2$ and then calculating the integral, but got nowhere. Any hints?  Thanks in advance! I am trying to put two subscripts at the left and right of a character. For example, something like:  _{t} p_{x}  where p is in the middle. How do you do this? How to import a jar in Eclipse Related:    The helpcenter has . It still links to the  for further details. Besides that, the question ban message .  Now that we have the helpcenter, could the following changes be implemented?   Link to the help center instead of the meta post Remove the link to the meta post from the help center, and instead incorporate its contents into the helpcenter itself I am constructing a thought experiment about a spinning object that is floating in intergalactic space.  I assume that this object is about the size of a planet so that it will have enough gravity so that a Foucault pendulum will work, although I'm not sure that this is necessary for the thought experiment.    I can easily determine that this object is spinning if I stand on the object and observe the galaxies around me rise and set.  Similarly, I can determine the axis of rotation.    For simplicity, I go to one of the poles of my object, and I set up my Foucault pendulum.  What will I see?  And why will I see it?  I assume that in intergalactic space, the gravity is very small (galaxies are very far away, and the mass of my object is very small compared to the mass of a galaxy) so that there will be little coupling between the gravitational field of my object and the gravitational field of the galaxies around me. I'm trying to figure out the best way to model the sphere-spiral here:    Is it possible to make this shape directly? Or do you have to make it as a cylinder and wrap it to a sphere? What is the point of the Red Thread/Tape around evidence or newspaper clippings? I have seen in a lot-lot of movies that the person who searches for a crime or another person creates a web of red tape or wire around the newspapers and evidence he acquires. I never guessed its usefulness and am therefore asking why do they do it?  Here is an illustration I created: I have a data frame in python where I need to find all categorical variables. Checking the type of the column doesn't always work because int type can also be categorical.   So I seek help in finding the right hypothesis test method to identify if a column is categorical or not.   I was trying below chi-square test but I am not sure if this is good enough  import numpy as np data = np.random.randint(0,5,100) import scipy.stats as ss ss.chisquare(data)   Please advise. How do you get the footer to stay at the bottom of a Web page? I have a simple 2-column layout with a footer that clears both the right and left div in my markup.  My problem is that I can't get the footer to stay at the bottom of the page in all browsers.  It works if the content pushes the footer down, but that's not always the case. Can you transfer a Windows 7 XP-mode VHD to a new PC without the need for reactivation? At the moment I am running Vista 32 bit and use VPC 2007 a lot. I have several Windows 2000 Virtual Machines set up with complex software configurations. For me, one of the advantages of working with VPCs is that a Virtual Machine can easily be copied to another PC (like a notebook). Also, when I move to a new PC, I can immediately work with my existing Virtual Machines without the need to reinstall all applications.   I am thinking about buying a new pc with Windows 7 64 bit pro, and using the XP-Mode Virtual Machine that comes with Windows 7. The advantages would be support for more memory (so being able to run more Virtual Machines simultaneously) and having a licenced XP Virtual Machine.  I would like to know if you can transfer a VHD that is based on the XP-mode VHD to another machine without Activation issues. The other machine will of course, also be running Windows 7. I would guess that this would not be a problem.   However, I did read about a guy who had to reinstall his Windows 7 because of a harddisk failure, and could not access his XP-mode VHD anymore because it demanded re-activation.  So can you transfer a Windows 7 XP-mode VHD to a new PC without the need for reactivation? Outlining in Blender? I am currently working on converting my logo vectored as a SVG to a 3D model using the 2.72 version. I ran into a problem with my logo because it contains a black stroke and an orange stroke (see both images below), but currently all I have is the interior color of each shape. I was wondering whether or not there was a simple way to add in these strokes, and how exactly to go about creating the strokes. Thank you in advance community! I really can't understand what Leonard Susskind means when he says in the video  that information is indestructible.  Is that information that is lost, through the increase of entropy really recoverable?   He himself said that entropy is hidden information. Then, although the information hidden has measurable effects, I think information lost in an irreversible process cannot be retrieved. However, Susskind's claim is quite the opposite. How does one understand the loss of information by an entropy increasing process, and its connection to the statement “information is indestructible”.  Black hole physics can be used in answers, but, as he proposes a general law of physics, I would prefer an answer not involving black holes. Evolutionarily speaking, why do humans have 46 chromosomes Claim: If $(x_\alpha)_{\alpha\in A}$ is a collection of real numbers $x_\alpha\in [0,\infty]$ such that $\sum_{\alpha\in A}x_\alpha&lt;\infty$, then $x_\alpha=0$ for all but at most countably many $\alpha\in A$ ($A$ need not be countable).  Proof: Let $\sum_{\alpha\in A}x_\alpha=M&lt;\infty$. Consider $S_n=\{\alpha\in A \mid x_\alpha&gt;1/n\}$.  Then $M\geq\sum_{\alpha\in S_n}x_\alpha&gt;\sum_{\alpha\in S_n}1/n=\frac{N}{n}$, where $N\in\mathbb{N}\cup\{\infty\}$ is the number of elements in $S_n$.  Thus $S_n$ has at most $Mn$ elements.   Hence $\{\alpha\in A \mid x_\alpha&gt;0\}=\bigcup_{n\in\mathbb{N}}S_n$ is countable as the countable union of finite sets. $\square$  First, is my proof correct? Second, are there more concise/elegant proofs? Adding values to a C# array Edge won't connect to internet, Skype for business won't screen-share, Windows Store won't update apps after Windows 10 version 1809 installed Most games build on other games' successes and many are downright clones. Where is the limit of imitation before legal issues come into play? Is it down to literary or graphic work like characters and storyline that cause legal problems, or can someone actually claim to own gameplay mechanics?  There are so many similar clone games out there that the rules are probably very slack or nonexistent, but I'd like to hear the views of more experienced developers / designers. Kelvin Source Terminal on MOSFETs On MOSFETs with multiple source pins, is the so-called &quot;Kelvin-connected source pin&quot; connected any different internally? Or is it just like every other source pin?   EDIT: Just to be clear, I am not asking what the Kelvin connection is for. I'm asking whether it is physically implemented any differently than the other source pins or whether it is just an extra regular, old source pin. formal proof from calulus Given $f:R \to R$, $f$ is differentiable on $R$ and $\lim_{x \to \infty}(f(x)-f(-x))=0$. I need to show that there is $x_0 \in R$ such that $f'(x_0)=0$  I am trying to prove it by contradiction .... so i assume there is no $x_0 \in R$ such that  $f'(x_0)=0$ this means this function is strictly monotonic because the derivative "respects" the mean value theorem so it cannot be negative and positive without passing a zero value ....  (from what i understand the derivative function of a continuous/differentiable functions isn't necessarily continuous but it still respect the mean value theorem i think it's called darboux's theorem )  now from $\lim_{x \to \infty}(f(x)-f(-x))=0$ ,we get $\lim_{x \to \infty}f(x)=\lim_{x \to \infty}f(-x)$  now the last step is the contradiction its very intuitive and even maybe obvious that we can't get $\lim_{x \to \infty}f(x)=\lim_{x \to \infty}f(-x)$ for strictly monotonic function but i cant think of a formal way to prove it .