Define a relation $R$ on $\mathbb{Z}\times(\mathbb{Z}\setminus\{0\})$ by $(a,b)R(x,y)\iff ay=bx.$ $a)$ Prove that $R$ is an equivalence relation. $b)$ Describe the equivalence classes corresponding to $R$.  I know that an equivalence relation satisfies transitivity, reflexivity, and symmetry. Also, I know that the Cartesian Product is $A\times B=\{(a,b):a\in A$ and $b\in B\}. $If I could get a hint for $a)$, that would be appreciated. For part $b)$, could someone explain what an equivalence class is? Please refrain from complete solutions. Thank you.
Let $r$ be a relation on $A \times A$ such that $(a,b) r (c,d) \iff ad = bc.$ How can I show that this relation is transitive, ie. $(a, b)r(c,d)$ and $(c,d)r(e, f) \implies (a,b)r(e,f)$?  I tried to say that $(a,b) r (c,d)$ means that $c=ka$ and $d=kb,$ $k$ a coefficient, such that $ad = bc \iff akb = bka,$ and going on from there, but I'm not sure this is valid for all values of $a, b, c,$ and $d$. How can I show the transitivity of the relation?
Why do we need to set swap space as twice big as our physical memory? When we want to setup a linux system, there is a common a suggestion to set the swap space  twice as big as your physical memory. I want to know why do we need this, and how it is this suggestion come to be? Why do my shortcut icons get broken on Windows 7? I have a strange problem. My icons seem to disappear randomly on my Windows 7. They were there a while back, now they ain't.  It happens often and quite randomly. Anyone know a fix? Carrying a handgun in other countries with a US concealed carry permit Getting CalloutException when testing Batchable class with Callouts and HttpCalloutMock How to concatenate multiple lines of output to one line? Is there a way to show Spotlight indexing status/progress in Yosemite? In previous versions of OS X, you could click the Spotlight icon in the menu bar to show the status of the Spotlight indexing, and it would show an estimate ("6 hours left"), along with a progress bar.  I've updated to Yosemite yesterday, and I noticed that Spotlight is not showing some expected results, i.e. files I know to be there, but it just doesn't find them if I type in their name and file type ("kind:excel vacation").  When I start typing in the Yosemite Spotlight window, it shows a small progress bar under the search box, but there's no estimate.    Is there another way to show the progress or an estimation on how long the indexing is going to take? Is it Possible to Disable Feed and Topics for an Object Completely at Profile Level / Page LayOuts $\phi \times \theta$ is faithful iff $(|Z(H)| ,|Z(K)|)=1$ for faithful characters $\phi \in Irr(H)$ and $\theta \in Irr(K)$ . Hartshorne Theorem 8.17 Are taglines & signatures disallowed? I recently had a number of my valid answers voted down because people objected to the fact that I added a tagline to the end. The answers were correct for the questions asked and the downvotes were only due to the tagline regarding hiring, which said:     We're hiring! Developers and QA in Washington, DC area (or looking to relocate) should send resumes to careers@example.com.   Are taglines and signatures disallowed?     For more information, see "" in the . Word for one who likes to share knowledge when asked There is already a similar post, that has an answer to my question, but it is quite short and I don't get it... So I am trying to show that  \begin{align} \{e_K |~ K \subset G~ conjugacy~ class \} \end{align} is a basis for $Z(\mathbb{Z}[G])$. Serre even writes "one immediately checks that the $e_K$ form a basis", but even reading the other post  I don't see that...  My approach to this is:  Be $x \in Z(\mathbb{Z}[G])$, which is equivalent to: $x = gxg^{-1}$ for all $g \in G$. Writing  $x = \sum\limits_{h \in G} \lambda_h h$, we get $x = \sum\limits_{h \in G} \lambda_h ghg^{-1}$. I don't really know how to go on at this point. I mean, I know that when $a,b \in K$ (same conjugacy class) you get $\sum\limits_{s \in K} \mu_s s = \mu\sum\limits_{s \in K} s$, since \begin{align} a = hbh^{-1} \Rightarrow \mu_a = \mu_b ~\forall a,b\in K \end{align} So all elements in the same conjugacy class have the same scalar.  But from here I don't really know how I can generate the $x = \sum\limits_{h \in G} \lambda_h ghg^{-1}$ with the $e_K$ ...  I would be really thankful, if someone could give me quite a basic proof that even I understand... Apparently recurrences like this cannot be solved with the Master Theorem:  $T(n) = 2T\left(\frac{n}{2}\right) + \frac{n}{\log(n)}$  Because $n^{\log_b(a)} = n^1$ is not a polynomial multiple of $f(n) = \frac{n}{\log(n)}$ since $\frac{n}{\frac{n}{\log(n)}} = \log(n)$, which is less than $n^\epsilon$ for any positive $\epsilon$.  But when is this really even an issue?   For example why wouldn't the time complexity be $n \log n$ using case 2 of the Master Theorem? Unrolling the recursion:  $$T(n) = n\sum_{k=1}^{\log_2(n)} \frac{1}{\log_2(2^k)}$$  $$T(n) = n\sum_{k=1}^{\log_2(n)} \frac{1}{k}$$  Doesn't this suggest $\Theta(n \log n)$ even though it supposedly cannot be done with Master Theorem? Can I gift games from my library? I understand that once a game is in the library it cannot be removed, only hidden.  My question is, can I gift an uninstalled copy of a game to a friend?  I understand that this may be impossible due to legal issues, and that once a game is in the library it stays. What can an app do with the "USE ACCOUNTS ON THE DEVICE" permission? Can such an app freely read my emails/calendar/message/docs/etc.?  The "USE ACCOUNTS ON THE DEVICE" permission is also known for developers as .  Example of such apps: , . $f:I=[\pi,2\pi]\to \mathbb{R}^2$ be given by $f(t)=(\cos t,\sin t)$  which of the following are necessarily correct?  $1$. $\exists t_0\in I$ such that $f'(t_0)=\frac{1}{\pi}(f(2\pi)-f(\pi))$  $2$. $\nexists t_0\in I$ such that $f'(t_0)=\frac{1}{\pi}(f(2\pi)-f(\pi))$  $3$.$\exists t_0\in I$ such that $\pi||f'(t_0)||\ge ||(f(2\pi)-f(\pi)||$  $4$. $f'(t)=(-\sin t,\cos t)\forall t$  $4$ is clearly true, $1$ is true by Mean Value Theorem as $f$  satisfies all the conditions of MVT? have no idea of $2$ and $3$, Thank you for help Is there a way I can avoid imperial ships? Whenever I am traveling on foot, imperial ships will fly by and drop down troops for me to fight. In a normal situation, this is no big deal and I will fight or flee to continue with my travels.  However this will also occur when I am doing a quest that requires me to locate and pick up items. This is an inconvenience as I cannot pick up items while in combat and I have to kill them before I can continue.  Is there a way I can "hide" and allow the ships to by pass my location? How to avoid perceived circularity when defining a formal language? Suppose we want to define a first-order language to do set theory (so we can formalize mathematics). One such construction can be found . What makes me uneasy about this definition is that words such as "set", "countable", "function", and "number" are used in somewhat non-trivial manners. For instance, behind the word "countable" rests an immense amount of mathematical knowledge: one needs the notion of a bijection, which requires functions and sets. One also needs the set of natural numbers (or something with equal cardinality), in order to say that countable sets have a bijection with the set of natural numbers.  Also, in set theory one uses the relation of belonging "$\in$". But relation seems to require the notion an ordered pair, which requires sets, whose properties are described using belonging...  I found the following in Kevin Klement's,  (pages 2-3).  "You have to use logic to study logic. There’s no getting away from it. However, I’m not going to bother stating all the logical rules that are valid in the metalanguage, since I’d need to do that in the metametalanguage, and that would just get me started on an infinite regress. The rule of thumb is: if it’s OK in the object language, it’s OK in the metalanguage too."  So it seems that, if one proves a fact about the object language, then one can also use it in the metalanguage. In the case of set theory, one may not start out knowing what sets really are, but after one proves some fact about them (e.g., that there are uncountable sets) then one implicitly "adds" this fact also to the metalanguage.  This seems like cheating: one is using the object language to conduct proofs regarding the metalanguage, when it should strictly be the other way round.  To give an example of avoiding circularity, consider the definition of the integers. We can define a binary relation $R\subseteq(\mathbf{N}\times\mathbf{N})\times(\mathbf{N}\times\mathbf{N})$, where for any $a,b,c,d\in\mathbf{N}$, $((a,b),(c,d))\in R$ iff $a+d=b+c$, and then defining $\mathbf{Z}:= \{[(a,b)]:a,b\in\mathbf{N}\}$, where $[a,b]=\{x\in \mathbf{N}\times\mathbf{N}: xR(a,b)\}$, as in  question or  on Wikipedia. In this definition if set theory and natural numbers are assumed, then there is no circularity because one did not depend on the notion of "subtraction" in defining the integers.  So my question is:     Question Is the definition of first-order logic circular?   If not, please explain why.   If the definitions are circular, is there an alternative definition which avoids the circularity?   Some thoughts:   Perhaps there is the distinction between what sets are (anything that obeys the axioms) and how sets are expressed (using a formal language). In other words, the notion of a set may not be circular, but to talk of sets using a formal language requires the notion of a set in a metalanguage. In foundational mathematics there also seems to be the idea of first defining something, and then coming back with better machinery to analyse that thing. For instance, one can define the natural numbers using the Peano axioms, then later come back to say that all structures satisfying the axioms are isomorphic. (I don't know any algebra, but that seems right.) Maybe sets, functions, etc., are too basic? Is it possible to avoid these terms when defining a formal language? How to calculate the variance of a partition of variables I'm running an experiment where I'm gathering (independent) samples in parallel, I compute the variance of each group of samples and now I want to combine then all to find the total variance of all the samples.   I'm having a hard time finding a derivation for this as I'm not sure of terminology. I think of it as a partition of one RV.   So I want to find $Var(X)$ from $Var(X_1)$, $Var(X_2)$, ..., and $Var(X_n)$, where $X$ = $[X_1, X_2, \dots, X_n]$.  EDIT: The partitions are not the same size/cardinality, but the sum of the partition sizes equal the number of samples in the overall sample set.  EDIT 2: There is a formula for a , but it only covers the case of a partition into two sets, not $n$ sets. An Open Interval and a Half-open Interval are not Homeomorphic