In many mathematical textbooks the definitions are given in the following form (considering even numbers as an example): &quot;A natural number n is called even iff it is a multiple of 2&quot;. Now as we know a definition should serve only as an abbrevation. That is I can use the following form for defining even numbers. &quot;Even number = a natural number that is a multiple of 2&quot;. Is there any difference between the above definitions? I was thinking that the first makes the definition more strict. For example, suppose that I state: &quot;$1.5$ is an even number&quot;. According to the first definition this statement is meaningless because the term &quot;even&quot; is defined for natural numbers. According to the second definition the statement should be false because $1.5$ isn't a natural number. Which of the above definitions is more appropriate and why? Edit I have read the question with the proposed answer before I asked this question. But there is a difference because I am asking what notation should I use and if there is a difference between equality sign and &quot;iff&quot; whereas the other question is about the difference between &quot;if&quot; and &quot;iff&quot;.
In some books the word "if" is used in definitions and it is not clear if they actually mean "iff" (i.e "if and only if"). I'd like to know if in mathematical literature in general "if" in definitions means "iff".  For example I am reading "Essential topology" and the following definition is written:     In a topological space $T$, a collection $B$ of open subsets of $T$ is said   to form a basis for the topology on $T$ if every open subset of $T$ can be written as a union of sets in $B$.   Should I assume the converse in such a case? Should I assume that given a basis $B$ for a topological space, every open set can be written as a union of sets in $B$?  This is just an example, I am not asking specifically about this sentence.
In point-set topology, one always learns about the : the topology on an infinite product $X = \prod_{i \in I} X_i$ generated by sets of the form $U = \prod_{i \in I} U_i$, where $U_i \subset X_i$ is open.  This seems naively like a "good" topology to use for $X$.  However, one quickly learns that this is not so; that the  is the natural one.  The box topology has many strange properties that make it a good source for counterexamples, but I am not aware of it having any other applications.  So I would like to know:     Are there examples of using the box topology to prove interesting "positive" statements?   Edit: And to pursue a comment of Jim Conant:     Are there "non-artificial" problems where the box topology arises naturally?   Edit: The title is perhaps too flippant.  I don't mean to minimize the obvious significance of the box topology as a counterexample.  However, for the purposes of this question I am interested in positive results.  I'm not looking to be convinced that counterexamples are useful; I know that they are. I am planning to visit the UK on a USA travel document but there’s nowhere I am able to find information about visas: I’m a Zimbabwean but using a USA refugee travel document do I need to apply for a UK visa before I travel or I can they just give them on arrival. Prove that $V = V_1 \oplus V_2$ in the following Let $V$ be a vector space over $\mathbb{C}$ such that $T^2=1_V$. Define  $V_1 = \left\{v\in V |\ T(v) = v \right\}$,  $\ V_2 = \left\{v\in V |\ T(v) = -v\right\}$  prove that $V = V_1 \oplus V_2$  For start, I've shown that $V_1\ ,V2$ are subspace of $V$. However, I'm stuck trying to show that $\forall \ v\in V,\ v = v_1 + v_2$ where $v_1 \in V1, v_2 \in V_2$.  I know at some point I have to make use of  $T^2=1_V$, but cant figure out how. Any help or insight is deeply appreciated. I remember reading it while in middle school in English in the United States, so the latest it would have come out was 1998. I don't remember if it was all one author, or even the same universe. These are the stories I remember:   A man finds a way to create a medicine that prevents radiation poisoning (or possibly makes someone fireproof... it's a little confused in my head) and learns that part of the problem is finding a market for it. I vividly remember the inventor telling someone to not "let the doorknob ream you" as it slams shut. A bout of graffiti artists leads to the city buying anti-graffiti paint that dissolves any other paint put on top of it. After learning the futility of trying to paint over it, the protagonist consults with a friend and learns that the anti-graffiti paint will eat away at the paint above it at a steady rate, and will even eat successive layers of the anti-graffiti paint. he uses this property to engineer an animation involving a harlequin thumbing this nose at the officials even as they try to wash it away. Some guys (I want to say a gang) go to a clinic for an illegal nanotech transformation. The viewpoint character gets turned into a woman while a friend gets turned into a dog. After roaming about in their new forms, they return to find the clinic being raided, meaning there's no one to turn them back. I think I remember something bad happening to the one transformed into a dog where the transformation continued, erasing his human thought patterns. There was also a scene where the viewpoint character submitted to being sexually assaulted by his former friends as if it was simply to be expected given his form.   I don't remember if the first or the second story was the actual first story of the collection... I want to say that those are the order that they showed up as in the book. XIV pages in page counts? Using the affine linearized polynomial $f(x)=x^p-x-b$ is irreducible over $\mathbb{F}_q$ if and only if $\operatorname{Tr}_{\mathbb{F}_q/\mathbb{F}_p} \neq 0$. But here $\mathbb{F}_3$ is not a field extension of $\mathbb{F}_5$.  Also we have that: Let $a \in \mathbb{F}_q$ and let $p$ be the characteristic of $\mathbb{F}_q$. Then the trinomial $x^P - x - a$ is irreducible in $\mathbb{F}_q[x]$ if and only if it has no root in $\mathbb{F}_q$. I've got some lenses of brand X, but I've just got a new camera of brand Y. How can I tell if it's possible to use my lenses on my camera? I accept that I may lose a lot of automatic functionality like autofocus, aperture control and metering when using an adapter to connect the two. SEO preference: does a site with WWW rank better than without WWW? After having read various articles on this site about WWW or Non-WWW website setup, ,  and , and after having setup a correct redirection rule in .Htaccess to at least eliminate duplicate contents (currently the non-www solution (e.g. http://example.com) I have a feeling that my site rank less since I changed this.   My confusion was strengthened by these two opponents:     &amp;     Could the change of www.example.com >> http://example.com have a factual impact on ranking?  say either is better or worse. Are they holding information back?  I am confused fellow folks...  As much as I would like to separate facts from fiction, I am aware that not all the good valid information is written. Not on google webmaster site and perhaps not elsewhere. If so it seems out of date!  Does or does not a WWW domain impact SEO/rank better than its NON-WWW version? (Given that you have setup a proper redirect to avoid duplicates and go with one) Printer service not found on port 515 For services list on my Ubuntu, Printer service not found on port 515.  nmap -sT -O localhost  PORT     STATE SERVICE 23/tcp   open  telnet 139/tcp  open  netbios-ssn 445/tcp  open  microsoft-ds 631/tcp  open  ipp 9050/tcp open  tor-socks Device type: general purpose Running: Linux 3.X OS CPE: cpe:/o:linux:linux_kernel:3 OS details: Linux 3.12 - 3.19, Linux 3.8 - 3.19       There is no Printer service on port 515. How to add Printer service on port 515?     nmap -v -p 515 127.0.0.1  Starting Nmap 7.01 ( https://nmap.org ) at 2016-11-19 13:03 IRST Initiating SYN Stealth Scan at 13:03 Scanning localhost (127.0.0.1) [1 port] Completed SYN Stealth Scan at 13:03, 0.21s elapsed (1 total ports) Nmap scan report for localhost (127.0.0.1) Host is up (0.000065s latency). PORT    STATE  SERVICE 515/tcp closed printer If a car hits a wall at a speed of $v$, and an identical car hits a wall at a speed of $2\,v$, with the kinetic energy equation $E_K = \frac{1}{2}mv^2$ (the speed goes from $v$ to $0$) the second car will lose 4 times the energy, and therefore cause 4 times the damage, but with the force equation $F = ma$ (the car hits the wall and decelerates), it will hit with 2 times the force, and cause 2 times the damage.  How can this be explained? Is there any way to automatically rename accessor/mutator when a variable they get/set gets refactored -> renamed (Eclipse 3.4)? How do you find a question's view counter? How do you see how many views a question has had when you're not in the newest category?  I've tried a word-for-word search for the question, but still can't see the view counter. Prove previsibility and $E[X_{T \wedge n}] \le E[X_{S \wedge n}]$ Sum of the series $\sum _{ r=1 }^{ \infty }{ \frac { { r }^{ 3 } }{ r! } } $ How do I prove that the sum of the series  $1+\frac { { 2 }^{ 3 } }{ 2! } +\frac { { 3 }^{ 3 } }{ 3! } +\frac { { 4 }^{ 3 } }{ 4! } +.... = 5e$ Why was a moved question deleted? Moderator to whom deletion is attributed claims they didn't do it A  appears to have been deleted, and deletion is attributed to a moderator,  deleted by Sathyajith Bhat♦ 12 hours ago , however,  (however, they did previously (unsuccessfully) tried migrating it to another site in the network where they're also a moderator, but migration was rejected by another moderator, it was then on hold on webapps, and then the hold was removed, which was supposed to have been the end of it).  Can someone explain what's going on here?  And restore the question that clearly wasn't meant to have been deleted, was a HNQ, and has 75 votes on the question, plus 74 votes on the top answer, and 56 on second answer; and 19 on third?  There's a lot of links to this question, too, and now they're all broken, making SE look like a community hostile to a popular question ontopic for the site it was on. As small as it may be, does every 'thing' have a gravitational pull?  That is, something with mass or energy. No matter how obsolete or negligible it may be, is it there? If so, how is it calculated? What does 'it' affect? Does a GFCI receptacle provide overcurrent protection? Does a GFCI receptacle provide over-current protection? (note that I'm not referring to GFCI circuit breakers as may be found in a service panel, nor to specialty devices such as an inline plug-in GFCI that may offer over-current protection)  I've checked some GFCI  (note, this link opens a PDF) but haven't found any that mention overcurrent protection, or provide trip curves as would be expected in a circuit breaker datasheet.  So, does a GFCI receptacle provide any inherent over-current protection or must it always be used with a circuit breaker for over-current protection?  This question was prompted by  where the answer suggests using a GFCI outlet as a circuit breaker. This seems like very big misconception of the protection offered by a GFCI, and is potentially dangerous if it leads someone to think that GFCI outlets are the same as circuit breakers. I resized the root partition with GParted, from 20 GB to 40 GB, but the filesystem still reports a size of 20 GB.  How can I use all 40 GB?    Update  The resized partition was an LVM Physical Volume (/dev/sda2), and the boot partition is ext4 (/dev/sda1). I know this is a really basic question, but I've just started with some basic C++ programming after coding a few projects with high-level languages.   Basically I have three questions:   Why use pointers over normal variables? When and where should I use pointers? How do you use pointers with arrays? I am new to salesforce. I dont have coding background. Please let me know bst way to learn apex coding?