So far I know that if $A \subset B$ then every element that is in $A$ will be also in $B$ but there will be at least one element in $B$ that is not in $A$  What about $A \subseteq B$?  site says that $A \subseteq B$ says that "$A$ is a subset of $B$. set $A$ is included in set $B$." But well, it's pretty much the same definition as $A \subset B$.  So my question is, $A \subset B$ and $A \subseteq B$, what is the difference?
If every element of set A ia an element of set B then we say that A is subset of B. A is said to be proper subset if A does not equal B.  My question is - whether the symbol we use to show subset is  same as that for proper subset? I have seen that A is subset of A itself is shown by the same symbol as that for the subset while when we write improper subset, we usually write equal to sign with symbol of subset. So, please help me to know this.
How do I use Linq to obtain a unique list of properties from a list of objects? I'm trying to use Linq to return a list of ids given a list of objects where the id is a property. I'd like to be able to do this without looping through each object and pulling out the unique ids that I find.  I have a list of objects of type MyClass and one of the properties of this class is an ID.  public class MyClass {   public int ID { get; set; } }   What I want to do is write a Linq query to return me a list of those Ids  How do I do that given an IList&lt;MyClass&gt; such that it returns an IEnumerable&lt;int&gt; of the ids?  I'm sure it must be possible to do it in one or two lines using Linq rather than looping through each item in the MyClass list and adding the unique values into a list.  Any help in creating an elegant solution would be much appreciated! Deleting a suspicious malware (imklaunchagent) in Mac I have a MacBook Pro (macOS Mojave) laptop and have recently noticed a suspicious process in Activity Monitor with this name : imklaunchagent    I checked this folder: /System/Library/Frameworks/InputMethodKit.framework/Versions/A/Resources/imklaunchagent and it seems imklaunchagent and its related files cannot be removed. I tried to remove it via Terminal and this message showed up: "Operation not permitted". However, in "Security and Privacy" of "System Preferences", I chose Terminal to have full disk access, still can't delete the file and the same message appears.  I installed the free version of Combo Cleaner and it detected other viruses, none of which are imklaunchagent. Is imklaunchagent a malware? How can it be deleted? How to move files to specific folders based on CSV entries? Adding coordinate system to layout in QGIS composer? I would like to know if there is a way to add a dynamic coordinate system field into a QGIS composer layout.  I can use an expression to add many items such as document name, file path, etc but I can;t seem to figure out how to add the coordinate system? How to write special set notation by hand? The foreach identifier and closures I am just curious, I use handlebar mounted lights, and since I have multiple bikes to switch, it's a lot of trouble changing the light from one to another. So, I am thinking of switching to helmet mounted lights.  Give me your point of view. Why or why not? Which is better suitable for which riding type? PHP considers null is equal to zero A result of Erdős on increasing multiplicative functions Erdős proved that if $f(n)$ is a monotone increasing function from the natural numbers to the positive reals, and $f(n)$ is completely multiplicative, then there exists some constant $C$ such that $f(n)=n^C$ for all $n$.  I have not been able to find a nice proof of this online and was hoping someone could provide a proof. Group theoretic meaning of natural isomorphisms between certain functors So imagine you have a group $G$ and we consider the set of group homomorphisms from $\mathbb{Z}$ to  $G$ specified by  $\forall g$ $\in G$ $\exists$   $\phi(1)=g$. Each of these homomorphisms is in 1-1 relation with the elements of $G$.  Consider groups for a moment in the context of category theory, viewing the groups as one object categories and thereof group homomorphism as functors. What would a natural isomorphism mean in group theory terms for G?  My thoughts are that the natural transformations equate to maps between homomorphisms and since the homomorphisms are in 1-1 relation with the elements of G, then a natural transformation is a map between group elements, i.e another element of the group. I'm not entirely convinced by this.   Anyway continuing my reasoning a natural isomorphism is a map between the homomorphisms that has an inverse, the thing is in relation to the elements of G this is every element? Furthermore I'm looking to find an equivalence class defined by this natural isomorphism, is this just a collection of the elements that are each others inverses?  Thanks in advance for any assistance. I have a correlation matrix of security returns whose determinant is zero. (This is a bit surprising since the sample correlation matrix and the corresponding covariance matrix should theoretically be positive definite.)  My hypothesis is that at least one security is linearly dependent on other securities. Is there a function in R that sequentially tests each column a matrix for linear dependence?  For example, one approach would be to build up a correlation matrix one security at a time and calculate the determinant at each step. When the determinant = 0 then stop as you have identified the security who is a linear combination of other securities.  Any other techniques to identify linear dependence in such a matrix are appreciated. I've installed a version of Ubuntu Server and although I'm fairly still fresh with Ubuntu, I'm trying to create a bash file to run a program in the background. More importantly, I'm trying to get a file nameto display correct and create it before I start running the program (I like my debug stuff).  However, here is my bash script:    #!/bin/bash  PATH="/var/log/servers/Game/"  FILENAME="startup"  EXT=".log"  DATE=$(date +%d-%m-%Y)  NEWFILE=${PATH}${FILENAME}_${DATE}${EXT}   I have verified that the file does exist in the PATH directory. However, the issue is with the date command. When I run the script bash -x game.sh this is what is displays.   + PATH=/var/log/servers/Game/  + FILENAME=startup  + EXT=.log  ++ date +%d-%m-%Y  game.sh: line 5: date: command not found  + DATE=  + NEWFILE=/var/log/servers/Game/startup_.log  + echo /var/log/servers/Game/startup_.log   I don't understand why it would say the command doesn't work, when I can go to the main console and run date and get a valid response.  Any help is greatly appreciated! Seeking QGIS tutorials and web resources? I wish to familiarize myself with the capabilities of QGIS up to and beyond QGIS version 3.0.   Can you recommend some good web resources and tutorials that would show me QGIS' capabilities?   I am interested in both using QGIS through its GUI and in developing applications using QGIS. What's the appropriate reaction to an answer that doesn't attribute the source it's copied from? How do I make log4j clear a log at startup? I installed Ubuntu 12.04 a month ago and am using it till now. I failed to notice that all this time there was no sound at all while running Ubuntu, even while playing a game in Wine. The weird thing is that only the startup sound comes when I log in (Indian/African drum tone), then comes the utter silence.  I tested both Digital Output (S/PDIF) and the speakers in the sound settings but can hear nothing.  Any help? I am using field image to insert image into article. The image can't show in article body.   How do I insert the image into article? MySQL convert date string to Unix timestamp Why does $a^n - b^n$ never divide $a^n + b^n$? How do I lerp between values that loop (such as hue or rotation)? I'm trying to make the joint rotate smoothly around the center of the canvas, toward the angle of mouse pointer. What I have works, but I want it to animate the shortest distance possible to get to the mouse angle. The problem occurs when the value loops around at the horizontal line (3.14 and -3.14). Mouseover that area to see how the direction switches and it takes the long way back around.  Relevant Code   // ease the current angle to the target angle joint.angle += ( joint.targetAngle - joint.angle ) * 0.1;  // get angle from joint to mouse var dx = e.clientX - joint.x,     dy = e.clientY - joint.y;   joint.targetAngle = Math.atan2( dy, dx );   How can I make it rotate the shortest distance, even "across the gap"?