a) Given a set $S$ with the $n$ elements, find the number of ordered pairs $(A,B)$ such that i) $A \subseteq B \subseteq S$ ii) $A \subset B \subseteq S$ b) Let n be a positive integer and S a set of size n. How many relations $\approx$ on S are there that are: i. an equivalence relation and a partial ordering? ii. an equivalence relation and a total ordering? Normally the topic that I ask about I have a general understanding of (Calculus) but I am having a really tough time with Discrete Maths... Any idea what I should do here?
Find total number of relations that are equivalence as well as partial order set. Assume set contains total $n$ elements.     My attempt:  As equivalence relation has property reflexive, symmetric and transitive and POSET has property reflexive, antisymmetric and transitive.  Symmetric and antisymmetric will cancle each other except reflexive relation and transitive relation can case antisymmetric, so it's also absent on resultant relation.  Therefore, all diagonal element should be present to satisfy conditions of equivalence as well as POSET.  So, it's independent of number of total number of element of set and answer will be $1$.     Can you explain in formal way, please?
Install packages from newer release without building (apt pinning) I need these packages with the latest upstream version backported for ubuntu/natty   libccid_1.4.5-0ubuntu1_amd64.deb libpcsclite1_1.8.1-0ubuntu1_amd64.deb libpcsclite-dev_1.8.1-0ubuntu1_amd64.deb libusb-1.0-0_1.0.9-0ubuntu1_amd64.deb libusb-1.0-0-dev_1.0.9-0ubuntu1_amd64.deb opensc_0.12.2-1ubuntu1ppa1~natty1_amd64.deb pcscd_1.8.1-0ubuntu1_amd64.deb pcsc-tools_1.4.18-0ubuntu1_amd64.deb   I tried to build them from source with update but it failed. I am not an expert at packaging and there are so many tutorials and recipes out there.  Do you know a good and up to date tutorial for backporting Ubuntu packages?  How do I get packages from other releases without having to build them myself? SF novel -- the only human on a space station full of aliens Some things I remember:   Pretty sure the author was female The novel had won some award or other The title was one word (I think) and it started with "I" (I think) It would have come out in the mid to late 90s or early 2000s It was a paperback or trade paperback, in English.  The book was originally written in English as far as I know. The cover, if I remember correctly, had an orangish background.   The plot involved a sole human representative to what was effectively an alien "United Nations" type of organization.  There were myriad types of aliens, many of whom were completely biologically incompatible.  Even the ones that could live in the same conditions generally found each other either repugnant or incomprehensible.  The main character ended up feeling incredibly lonely and was "set up" with a superficially similar humanoid female, but found their pheromones were incompatible and it was still difficult to communicate. Is it possible to enable System Protection from the command prompt? What's wrong if I fit the auto-regression with OLS? I am doing auto-regress by usual linear regression package. e.g.  $y_t=φx+ε_t$ with $x =y_{t-1}$  My reason is that, Auto-regression does assumes iid errors, same for linear regression. Linear Regression doesn't have assumption on independent variables. What's different is merely that the independent variables is replaced with lagged dependent variable y.  But I see AR always is fitted by specific methods, I am afraid I'm doing wrong. I'm getting a NoSuchMethodError error when running my Java program.  What's wrong and how do I fix it? Libvirt: how to pass qemu command line args? I have the current libvirt 1.2.20 i want to pass to my machine create with virt-manager some qemu flags. I edited the file and add    &lt;/devices&gt;   &lt;qemu:commandline&gt;           &lt;qemu:arg value='-L pc-bios'/&gt;   &lt;/qemu:commandline&gt; &lt;/domain&gt;   But doesn't work i also tried to edit from virsh  virsh edit domain   and put   &lt;qemu:commandline&gt;           &lt;qemu:arg value='-L pc-bios'/&gt;   &lt;/qemu:commandline&gt;   But give me this error  errore: XML document failed to validate against schema: Unable to validate doc against /usr/share/libvirt/schemas/domain.rng Element domain has extra content: qemu:commandline  Failed. Try again? [y,n,i,f,?]:    How to pass correctly qemu command line? Thanks How can I launch a GameObject at a target if I am given everything except for its launch angle? I am trying to launch an object at a target, given its position, its target position, the launch speed, and the gravity.  I am following this formula from :  $$ \theta = arctan \bigg( \frac{v^2 \pm \sqrt{v^4-g(gx^2 + 2yv^2)}}{gx} \bigg) $$  I have simplified the code to the best of my ability, but I still cannot consistently hit the target.  I am only considering the taller trajectory, of the two available from the +- choice in the formula.  Does anyone know what I am doing wrong?  using UnityEngine;  public class Launcher : MonoBehaviour {     public float speed = 10.0f;      void Start()     {         Launch(GameObject.Find("Target").transform);     }      public void Launch(Transform target)     {         float angle = GetAngle(transform.position, target.position, speed, -Physics2D.gravity.y);         var forceToAdd = new Vector2(Mathf.Cos(angle), Mathf.Sin(angle)) * speed;         GetComponent&lt;Rigidbody2D&gt;().AddForce(forceToAdd, ForceMode2D.Impulse);     }      private float GetAngle(Vector2 origin, Vector2 destination, float speed, float gravity)     {         float angle = 0.0f;          //Labeling variables to match formula         float x = Mathf.Abs(destination.x - origin.x);         float y = Mathf.Abs(destination.y - origin.y);         float v = speed;         float g = gravity;          //Formula seen above         float valueToBeSquareRooted = Mathf.Pow(v, 4) - g * (g * Mathf.Pow(x, 2) + 2 * y * Mathf.Pow(v, 2));         if (valueToBeSquareRooted &gt;= 0)         {             angle = Mathf.Atan((Mathf.Pow(v, 2) + Mathf.Sqrt(valueToBeSquareRooted)) / g * x);         }         else         {             //Destination is out of range         }          return angle;     } } What are neutral density filters and how do I use them to create long exposures in daylight? I've seen a few related posts on the site but nothing specifically on this point:  I'd like to be able to take long exposure photos during daylight (like the examples in ). I'm told that you can do this by using a neutral density filter. So my question is: what is a neutral density filter exactly, and how is it used to achieve that smooth long exposure effect? Assume: $p$ is a prime that satisfies $p \equiv 3 \pmod 4$  Show: $x^{2} \equiv -1 \pmod p$ has no solutions $\forall x \in \mathbb{Z}$.  I know this problem has something to do with Fermat's Little Theorem, that $a^{p-1} \equiv 1\pmod p$. I tried to do a proof by contradiction, assuming the conclusion and showing some contradiction but just ran into a wall. Any help would be greatly appreciated. Density of sum of two independent uniform random variables on $[0,1]$ I am trying to understand an example from my textbook.  Let's say $Z = X + Y$, where $X$ and $Y$ are independent uniform random variables with range $[0,1]$. Then the PDF is $$f(z) = \begin{cases} z &amp; \text{for $0 &lt; z &lt; 1$} \\ 2-z &amp; \text{for $1 \le z &lt; 2$} \\ 0 &amp; \text{otherwise.} \end{cases}$$  How was this PDF obtained?   Thanks Need hint: Determine whether the series $\sum_{n=1}^{\infty}(\frac{n}{n+1})^{n^2}$ is convergent or divergent. How do I enable the "multiverse" repository? I keep seeing places refer to the "multiverse" repository as a place I can get software, how can I enable this repository? Please specify how to do this graphically and via command line. How to find Windows 7 File Transfer History? Is it possible to find out which files are transferred to an external usb flash drive in certain period of time?   To be more clear. If I/someone have a flash drive and take some data from me. How do I know which files are copied? Finding out file copy hostory could be also a work around. I am currently working to design a flash virtual tour, but I know little about how the photographs are taken.  Ideally I'd like to be able to take a set of 6 shots from a single focal point, and be able to align them seamlessly into a cube form. The other solution is to take a special fisheye image, and stretch it dynamically in flash.  I've seen examples of both as virtual tours, and ones formatted as a rotating cube are more responsive and have a higher image quality. The airbus and Cruden Homes are definitely in the cube format, I'm not entirely certain which format the New York VT uses:        I have a set of questions, the first is the most important, the others may become separate questions if I don't get an answer here:   How are those photos taken? Is there a specific mount that can be used? How long does it take to set up? How much post-processing is necessary on the images? Is there a fast way (without looping over all edges/faces/verts) using ops.mesh or bmesh to get the lists of indices associated with mesh islands as shown below? These islands are part of one mesh but are disjoint from each other.    I know how to select the boundaries by running the operator Select boundary Loop or calling bpy.ops.mesh.region_to_loop() but can't find any API calls to return the vert/edge/face indices associated with the individual islands. The model used to open the store is incompatible with the one used to create the store How to validate an email address in JavaScript Is there a regular expression to validate an email address in JavaScript? Segmentation fault error occurs for a simple function I have coded I accidentally closed the window for the verfication code I wanted to create an account for Ubuntu One, but I got impatient when the email with the verification code didn't arrive... I thought I might have typed the wrong address, closed the window and tried again. it told me the address already existed and I just got the code. But since I closed the window I missed it. Is there any way to recover the code? Validation Error less than training error? I found two questions  and  about this issue but there is no obvious answer or explanation yet.I enforce the same problem where the validation error is less than training error in my Convolution Neural Network. What does that mean?