S→∃x.Q(x) ⊢ ∃x.(S→Q(x)): How to prove the validity of this sequent in predicate logic?
How to prove this sequent using natural deduction?
"There Is"/"There are" depends on plurality of the first list element or not? Yesterday, Asaf Karagila's  sparked an extensive discussion on ways of proving that all countable ordinals are embeddable in $\Bbb Q$, and whether particular solutions to this use the Axiom of Choice.  Since the arguments used to prove this can be shady on their use of Choice when formulated tersely, fully fleshed out answers are appreciated.  To be concrete, this question asks for proofs of the following fact:     All countable ordinals $\alpha &lt; \omega_1$ are order-embeddable in the rationals $\Bbb Q$.   Especially answers avoiding the Axiom of Choice are appreciated.    Of relevance and related interest is  (but it's not a duplicate, since that question asks for explicit embeddings). Naive visualization of space-time curvature Drop or change automatic migration-rejection based on non-existing tags I believe this feature creates more issues than it solves and thus would like to see it gone. (This is an out-growth of , which I repost here as it concerns several/all sites.)  The main reason why I think so is that there can be numerous reasons other than the subject being off-topic why some tag does not exist on some site, for one thing it could be a spelling, typesetting, or other formalistic reason. For example, for the two sites I know best MathOverflow and Mathematics hardly any of the most popular tags have a strictly matching tag on Mathematics but only because on MO there is the convention that certain tags are formatted in a specific way, which is not in place on Mathematics (for example nt.number-theory vs number-theory).  On top of that a user choosing the wrong site in the first place is likely not too well aware of the tags of the site they are posting too so that the question might be poorly tagged in the first place.         There is an additional detail I do not understand (it is obsolete if the actual request should be followed but in case it is not):   Even if on the source site a tag has a synonym and that synonym exists as a tag on the target site the migration is rejected. I do not understand the logic behind this at all. Possibly this is just an oversight.   I could also think of some other ways to improve the process while keeping the idea, but since I believe that it is best dropped I do not (yet) elaborate on this. Prove that $$\nabla_{\mathrm X} \mbox{tr} (\mathrm A \mathrm X^{-1} \mathrm B) = - \mathrm X^{-\top} \mathrm A^\top \mathrm B^\top \mathrm X^{-\top}$$   My proof is below. I am interested in other proofs.    My proof  Let   $$f (\mathrm X) := \mbox{tr} (\mathrm A \mathrm X^{-1} \mathrm B)$$  Hence,  $$\begin{array}{rl} f (\mathrm X + h \mathrm V) &amp;= \mbox{tr} (\mathrm A (\mathrm X + h \mathrm V)^{-1} \mathrm B)\\ &amp;= \mbox{tr} (\mathrm A (\mathrm X ( \mathrm I + h \mathrm X^{-1} \mathrm V))^{-1} \mathrm B)\\ &amp;= \mbox{tr} (\mathrm A ( \mathrm I + h \mathrm X^{-1} \mathrm V)^{-1} \mathrm X^{-1} \mathrm B)\\ &amp;= \mbox{tr} (\mathrm A ( \mathrm I - h \mathrm X^{-1} \mathrm V + O (h^2)) \mathrm X^{-1} \mathrm B)\\ &amp;= \mbox{tr} (\mathrm A \mathrm X^{-1} \mathrm B) - h \, \mbox{tr} ( \mathrm A \mathrm X^{-1} \mathrm V \mathrm X^{-1} \mathrm B) + O (h^2)\\ &amp;= f (\mathrm X) - h \, \mbox{tr} ( \mathrm X^{-1} \mathrm B \mathrm A \mathrm X^{-1} \mathrm V ) + O (h^2)\end{array}$$  Thus, the directional derivative of $f$ in the direction of $\mathrm V$ at $\mathrm X$ is  $$D_{\mathrm V} f (\mathrm X) = - \mbox{tr} ( \mathrm X^{-1} \mathrm B \mathrm A \mathrm X^{-1} \mathrm V ) = - \mbox{tr} ( (\mathrm X^{-\top} \mathrm A^\top \mathrm B^\top \mathrm X^{-\top})^\top \mathrm V ) = - \langle \mathrm X^{-\top} \mathrm A^\top \mathrm B^\top \mathrm X^{-\top}, \mathrm V \rangle$$  and, lastly,  $$\nabla_{\mathrm X} \mbox{tr} (\mathrm A \mathrm X^{-1} \mathrm B) = - \mathrm X^{-\top} \mathrm A^\top \mathrm B^\top \mathrm X^{-\top}$$ On the proof that the product of diagonal matrices is a diagonal matrix? Let $A [a_{i,j}]$  and $B =[b_{ij}]$   be two diagonal matrices in   $M_n(F)$. Let $i \ne j \in \left\{1, \ldots, n \right\}$. Using the   product rule, we obtain      $$(AB)_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$$      We claim that $a_{ik}b_{kj} =0$ for all $k \in \left\{1, \ldots, n \right\}$ thus $(AB)_{ij}=0$ for all $i \ne j$ and $AB$ is diagonal. To    prove the claim, note that since $i \ne j$ , we have $i \ne k$ or $    k\ne j$. Thus either $a_{ik} = 0$ (since $A$ is diagonal) or $b_{kj} =  0$ (since $B$ is diagonal), thus in all cases $a_{ik}b_{kj}=0$ and the   claim is proved.   Where does the bit "we have $i \ne k$ or $k\ne j$" come from? I understand that since $i \ne j$, and $k$ and $j$ run over the same set of natural numbers, we also have $i \ne k$. But the "or $k \ne j$" baffles me. I understand that $k \ne j$ implies that we can have $i=k$. But that's from the other direction ($\Longleftarrow).$  Also, when it says $a_{ik}=0$ or $b_{kj}=0$ it's talking about when $i \ne k$ or $k \ne j$ right? Many thanks. How to start X windows inside "Bash on Ubuntu on Windows" After installing the new feature (still "Beta") called  that is a feature subset of Windows 10 Anniversary edition, I get the following message when trying to start X-Windows using startx :    What do I need to tweak / configure in order for X-Windows to launch?  I noticed that xauth was already installed, however I had to install the rest of the framework as follows :  apt-get install xorg openbox 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. Flight reservation for Schengen visa One of the necessary documents for a Schengen visa for non EU citizens is proof of your flights, but it doesn't state the actual (e-)tickets (for which I already paid), or the reservation of flights.  So my question is, which option would be more acceptable (if I can provide any of them) to provide printed tickets to an EU member embassy or reservation of flights (reservation does not expect payment)? If electrons were positive and protons were negative, would life be different? This was a question on a worksheet during my first week in a class on Electromagnetism. The answer is essentially:     No. Life would be no different if electrons were positively charged and protons were negatively charged. Opposite charges would still attract, and like charges would still repel. The designation of charges as positive and negative is merely a definition.   But how would we have negative charges within the nucleus? Taking a look at the , it seems that down quarks are required in this new "negative proton" to help with creating the pion to "transmit a residual part of the strong force even between colorless hadrons".  I tried to set out and try to find a particle with (-1) charge with a down quark (by going through this ) but all the particles are not stable with the exception of two unknown:   Bottom Xi Baryon Double Bottom Xi Baryon   Assuming one of the above are stable, could the strong force act between one of them and a neutron, implying its alright to have a negatively charged nucleus?  Edit: If it wasn't clear, my question concerned whether or not there existed stable baryons of $-1$ charge with down quarks that could "replace" the proton. Full URL not working with php include 2015 13" MacBook Pro Trackpad and Keyboard stop working Decorate Tables You have $r$ red, $g$ green and $b$ blue balloons. To decorate a single table for the banquet you need exactly three balloons. Three balloons attached to some table shouldn't have the same color. What maximum number $t$ of tables can be decorated if we know number of balloons of each color?  I know the answer will at least be $\min(r,g,b)$. Also we need to add tables with $2$ reds, $2$ blues or $2$ greens. I cannot think of the formula for it Wouldn't multiple filters in a convolutional layer learn the same parameter during training? Based from what I have learned, we use multiple filters in a Conv Layer of a CNN to learn different feature detectors. But since these filters are applied similarly (i.e. slided and multiplied to regions of the input), wouldn't they just learn the same parameters during training? Hence the use of multiple filters would be redundant? Div's class vs id When using divs when is it best to use a class vs id?   Is it best to use class, on say font variant or elements within the html? Then use id for the structure/containers?   This is something I've always been a little uncertain on, any help would  be great. Dell Latitude E7270 $$\displaystyle\int \:\sin^n\left(x\right)\cos^m\left(x\right)\mathrm  dx=\frac{\sin^{n+1}x\cos^{m-1}x}{m+n}+\frac{m-1}{m+n}\int \:\sin^nx\cos^{m-2}x\,\mathrm dx$$  I have been trying to solve for over a week now can someone please help me. How much money do I need to show as proof of support when applying for a UK Standard Visitor visa? I've had a promising Skype interview with a great professor from a top-level university in US. Our research interests are closely related and he encouraged me to apply BUT he says that all decisions are finally made by admission committee there.  The question is "Is a professor in such a high-level university able to affect admission committee if he really wants a student?" or "Does admission committee decision-making process become easier if a professor really wants an applicant?"  The main point is the difference between my transcript (which is a real catastrophe!) and my research background which is pretty good so the professor's decision might be different from admission committee. Access is denied because the NTFS volume is already exclusively opened