When reading books I pay attention that some of them giving a definition to some notion use "if" but others "if and only if". For example,   A set is called empty if it has no elements A set is called empty if and only if it has no elements   or    A set is a subset of another if all elements of the former are in the latter A set is a subset of another if and only if all elements of the first one are in the second one   As these statements involve naming of something, I can't fully understand which one should be correct. Of course "if and only if" seems reasonable for me and formal, but what about using just "if"?  If antecedent is true, then by a statement we would name it as it is described in the statement. From other side, given we call it with the name that coincides with the statement (consequent part holds), can we say its properties (antecedent part) hold? From my experience of reading books it seems true, but doesn't fully make sense to me.
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.
I wonder if there is somewhere an explanation for the parameters that can be set when running the Algorithm "Join Attributes by Location" in QGIS 2.18.  I had a look here:       but there is no explanation. Any hints? Given a DateTime object, how do I get an ISO 8601 date in string format? LWC to display salesforce objects as a picklist onclick I have a requirement where I need to display list of salesforce objects as a picklist when a radio button is clicked.  Being new to LWC development, I could not figure out what is the right approach to do so. I have come across  from Salesforce articles where we can display radio buttons but as per my scenario, i need to display list of all salesforce objects as picklist when a radio button option is selected.  Can anyone please suggest the right approach to do so. I want to draw a table and i used the following code  \begin{table}[htbp]  \centering \caption{Ship simulator required inputs} \begin{tabular}{|c|c|} \hline L_{PP},  L_{WL}, L_{OS}, B, T, T_{a}, T_{f}, C_{B}, D_{P}, `\\` Bulbous bow, Number of bosses, brackets, thrusters. &amp; ship particulars `\\` \hline Required speed or propeller rpm, ship course angle &amp; Mission data \\ \hline BN, weather angle, water temperature &amp; Environment \\  \hline \end{tabular}% \label{inputs}% \end{table}%`   the result of this is as shown in figure    How can i complete the right border of the table ??? I used \\ to break the line of first row Can I boil milk unattended (and still keep the pan clean)? Minecraft directory doesn't have a 'Resource Packs' folder? I go into my .minecraft folder and there's Assets and all the other files and folders, but no resource packs folder?  If I create one manually, and place the textures inside, they don't show up in-game.  I'm on a Mac. Prove: The positive integers cannot be partitioned into arithmetic sequences (using Complex Analysis) This is a pretty simple question, at least it seems like it should be, about sudo permissions in Linux.  There are a lot of times when I just want to append something to /etc/hosts or a similar file but end up not being able to because both &gt; and &gt;&gt; are not allowed, even with root.  Is there someway to make this work without having to su or sudo su into root? I have the following diagram:    However, this looks a bit ugly, I would like a version that looks a little bit better. For example, size the last arrow such that the T_\phi is moved to the right.  This is the code which I am using to do this:  \begin{align*}   \xi \colon &amp;\lb \to \mathcal{L}(\hh)\\   &amp;\phi \mapsto T_\phi. \end{align*}   Where \lb and \hh just give the names for those spaces. Does anybody have a suggestion how I would improve the look of this function definition? I have three displays, a 4K display flanked by two 1080p displays. Physically, these monitors are roughly the same size. Furthermore, Windows 10 handles the DPI scaling of the 4K monitor separately from the 1080p monitors just fine. So while it appears as such to Windows    the real layout is three monitors of equal size side by side.  The problem is that the mouse keeps getting caught on the edges of display 2 and three when dragging from display 1. I would like the mouse to continue onto display 2 or 3 regardless of which part of the left or right edge of display 1 the mouse touches.  This question has been asked before (e.g. ) and a common answer is to suggest that this behaviour is intentional on the basis that mouse movement should reflect different size displays. Such an answer is not appropriate as it doesn't consider DPI and scaling. Assume that $0 &lt; a_n \leq 1$ and that $\sum a_n=\infty$. Is it true that  $$ \sum_{n \geq 1} \frac{a_n}{(a_1+\ldots+a_n)^2} &lt; \infty $$ ?  I think it is but I can't prove it. Of course if $a_n \geq \varepsilon$ for some $\varepsilon &gt; 0$ this is obvious.  Any idea? Thanks. Do I need a Turkey transit visa for a layover? Expression Versus Statement I'm asking with regards to c#, but I assume its the same in most other languages.  Does anyone have a good definition of expressions and statements and what the differences are? Everybody who is using Lightroom probably has seen this. After importing RAW files, Lightroom shows a preview, but contrast and color of the preview change after a few short seconds to a slightly different view.  For some users the change is small, for others the change is huge. Colors are different (read: wrong), contrast is off.  Rumor says this is happening only (more?) for Nikon users.  What is Lightroom doing here?  I'd guess that for the first preview, LR is showing the JPG preview file, which is embedded in the RAW file. Then after a moment, LR renders the RAW file itself to show it's own interpretation.  But why is this interpretation so different from the embedded JPG? I need help running python in ubuntu  I can run python without a problem in my terminal but when I create a .py file I can't get it to run. Deleting my own question Why was my "shopping list" question closed? My question was closed for being a "shopping list question", even though it was very much related to the topic of the site. Why? I have heard that there are some Pokémon that are region specific.  Could someone please clarify which Pokémon are region specific and in which regions that they can be found in? This is not a duplicate question because I am looking for an explanation directed to a general audience as to the mistakes (if any) in  (reproduced below). (Numberphile is a YouTube channel devoted to pop-math and this particular video has garnered over 3m views.)   By general audience, I mean the same sort of audience as the millions who watch Numberphile. (Which would mean, ideally, making little or no mention of things that a general audience will never have heard of - e.g. Riemann zeta functions, analytic continuations, Casimir forces; and avoiding tactics like appealing to the fact that physicists and other clever people use it in string theory, so therefore it must be correct.)  Numberphile's Proof.  The proof proceeds by evaluating each of the following:  $S_1 = 1 - 1 + 1 - 1 + 1 - 1 + \ldots$   $S_2 = 1 - 2 + 3 - 4 + \ldots $   $S = 1 + 2 + 3 + 4 + \ldots $   "Now the first one is really easy to evaluate ... You stop this at any point. If you stop it at an odd point, you're going to get the answer $1$. If you stop it at an even point, you get the answer $0$. Clearly, that's obvious, right? ... So what number are we going to attach to this infinite sum? Do we stop at an odd or an even point? We don't know, so we take the average of the two. So the answer's a half."   Next:  $S_2 \ \ = 1 - 2 + 3 - 4 + \cdots$  $S_2 \ \ = \ \ \ \ \ \ \ 1 - 2 + 3 - 4 + \cdots$  Adding the above two lines, we get:  $2S_2 = 1 - 1 + 1 - 1 + \cdots$  Therefore, $2S_2=S_1=\frac{1}{2}$ and so $S_2=\frac{1}{4}$.  Finally, take  \begin{align} S - S_2 &amp; = 1 + 2 + 3 + 4 + \cdots           \\ &amp; - (1 - 2 + 3 - 4 + \cdots)          \\ &amp; =   0 + 4 + 0 + 8 + \cdots           \\ &amp; = 4 + 8 + 12 + \cdots           \\ &amp; = 4S \end{align}  Hence $-S_2=3S$ or $-\frac{1}{4}=3S$.  And so $S=-\frac{1}{12}$. $\blacksquare$ How to prove that r is a double root if and only if it is a root of a polynomial and of its derivative.