Trvially, the empty set $\emptyset$ is a subset of every set $V$(say). Does it mean that $\emptyset$ is also a subset of every subset of $V$? i.e , if $A$ is an arbitrary subset of $V$, then will it be right to say that $\emptyset$ $\subseteq A$?
This problem is from Discrete Mathematics and its Applications  My question is on 9b. I know that the sign represents an element is a member of. (from book)  I know that the O with a slash across it is the empty set which &quot;is a special set that has no elements&quot;. From , I got that the empty set is a subset of all sets, meaning that every member of the empty set(nothing) is also a member of any other set. Based on all of this, for 9b, would {0} contain the empty set because it fundamentally has the elements that consist of the empty set(nothing) or does it physically have to have the empty set?
Getting coordinates by clicking on QGIS Canvas with PyQGIS? Derive equation for a cantilever in SHM How to include custom non-system folder under This PC in Windows 8.1? I've recently upgraded from Windows 8 to 8.1 and noticed the restructured This PC folder, formerly Computer.   What I want to achieve is to include custom folder under "This PC".    What I've found after googling is that what is displayed under "This PC" is stored in registries under   HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Explorer\MyComputer\NameSpace  However those folders (such as Desktop, Documents, etc) are referenced via CLSID and the folder I want to include in the list does not have CLSID. Verify is tautology by using logical equivalence Verify is tautology by using logical equivalence:  $$((p ∨ q) ∧ (p → r) ∧ (∼ r)) → q$$  by showing what laws are using, step by step explanation needed.  Logical equivalence: de morgan's laws, commutativity, associativity, double negative, distribution, idempotent, absorption, identity, universal bound, negation  Rules of inference: modus ponens, modus tollens, generalisation, specialisation, conjunction, ellimination, transivity, proof by division into cases, contradiction  What I got so far:  I need to prove   $$((p ∨ q) ∧ (p → r) ∧ (∼ r)) → q ≡ \text{tautology}$$ Since $p → r$ same as $∼ p ∨ r$: $$((p ∨ q) ∧ (p → r) ∧ (∼ r)) → q ≡ ((p ∨ q) ∧ (∼ p ∨ r ) ∧ (∼ r)) → q$$  I am stuck here The question is from Axler's "Linear Algebra Done Right", which I'm using for self-study.  We are given a linear operator $T$ over a finite dimensional vector space $V$. We have to show that $T$ is a scalar multiple of the identity iff $\forall S \in {\cal L}(V), TS = ST$. Here, ${\cal L}(V)$ denotes the set of all linear operators over $V$.  One direction is easy to prove. If $T$ is a scalar multiple of the identity, then there exists a scalar $a$ such that $Tv = av$, $\forall v \in V$. Hence, given an arbitrary vector $w$, $$TS(w) = T(Sw) = a(Sw) = S(aw) = S(Tw) = ST(w)$$ where the third equality is possible because $S$ is a linear operator. Then, it follows that $TS = ST$, as required.  I am, however, at a loss as to how to tackle the other direction. I thought that a proof by contradiction,  ultimately constructing a linear operator $S$ for which $TS \neq ST$, might be the way to go, but haven't made much progress.   Thanks in advance! PS4 Controller not connecting to PS4 and won't reset? ''More than one row returned by a subquery used as an expression'' postgres error Very specific paragraph shape First, … Secondly, … &c Take it for granted that enumerations have secondly, …; thirdly, …; finally, … Is there any reason, except tradition, to prefer the traditional first to firstly? Functions satisfying $f:\mathbb{N}\rightarrow\ \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$ Find all functions $f$ such that $f:\mathbb{N}\rightarrow\  \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$   Let us plug in $n=1$  $f(f(1))+f(2)=3$  Since the function is from $\mathbb{N}$ to $\mathbb{N}$, $f(2)$ can only take the values $1,2$. Now we divide the problem into cases.  Case-1: $f(f(1))=2,f(2)=1$  We can assume that $f(1)=c$ for the time being. Then plugging in $n=3$ and using $f(2)=1$ gives $$f(3)=4-c$$ and again since the range of the function is positive integers,then $4-c$ has to be positive and hence $c$ belongs to  {$1,2,3$}. Now, $$f(1)=c$$$$\implies f(f(1))=f(c)$$$$\implies 2=f(c)$$ by the assumption that $f(f(1))=2$ . Now,since $c$ can only take the values $1,2,3$,we start treating cases. If $c=1$,$$f(c)=2$$$$\implies f(1)=2$$ but we know from the deinition of $c$ that $f(1)=c=1$,a contradiction.If $c=2$,then $2=f(c)=f(2)$ but $f(2)=1$ by assumption. Finally,if $c=3$ $$2=f(c)=f(3)$$ but $$f(3)=4-c=4-3=1$$ which is once again a contradiction. Therefore there are no such functions in this case.  Case-2: $f(f(1))=1,f(2)=2$  Again assuming $f(1)=c$ and using $f(n)\le n$ along with plugging $n=c-1$ will give us that $f(1)=1$ and then it is easy to prove that such a function exists by recursion. I can only give a "sort of recursive" way to define the function. Here it goes $$f:\mathbb{N}\rightarrow \mathbb{N}$$$$f(1)=1$$$$f(n)=n+1-f(f(n-1))$$  But this case is harder to deal with.Some help will be appreciated. How to make the BibTeX reference list visible in a LaTeX document Im very new at LaTeX and I would appreciate some help solving this problem. I have created a BibTeX file containing all references to a document. In this text file BibTeX items are inserted and no other text are written, for instance @article{...}. When I call for this bib file from the main.tex file I use the commands:  \bibliographystyle{plain} \bibliography{thebibliography}   Now to my question. All i get when I compile this main.tex is the title (bibliography) for this chapter, otherwise the textfield is blank. Is this error occurring because I should write anything in the bib file or have I missed any lines in the main.tex file. Windows 10 thinks it's up to date, but it's not Expected value of a non-negative random variable is that function must be constant under the following conditions I'm talking about complex function   $f$ is analytic function on a region $D$ that include the point $z=0$. for every $n\in N$ such $\frac{1}{n}$ is in $D$ , the function follows this condition $|f(\frac{1}{n})| \leq \dfrac{1}{2^n}$   is that function must be constant ? Is it possible to get the data off of the hard drive of a PlayStation 3 that has crashed? I have already voided the warranty so I am not worried about that and I have access to quite a few tools.   I have a new PlayStation and a few saves that were not backed up. I am not worried about them but if it is possible I would love to get it moved over.   The old PlayStation will not turn on at all and I think I am going to have to pull the drive. My main concern is if I am able to transfer the files without running into any issues. I have heard that some of the stuff is specific to each PlayStation, is that true? App Store is the wrong language are used by the Empire in a manner that is tactically identical to how a modern day army would use a tank, in that they are used as shock armour with powerful main guns which will punch through entrenched positions. (No combined arms tactics were used in the Battle of Hoth, but that's a different question altogether)  However, AT-AT walkers have a number of (rather lethal) drawbacks and no advantages that I can see compared to tanks.    They have footpads in place of treads, which greatly increase their ground pressure, and which will cause them to easily sink as compared to treads They have a high center of gravity and can be easily tripped, which was exploited in the Battle of Hoth. (granted that this was canonically presented as an oversight, albeit a rather stupid one, since most martial arts strongly emphasise lowering your CG.) They are slower than a modern main battle tank despite their long legs, which one would normally expect to grant additional mobility (max speed 60km/h on flat terrain, compared to 72km/h on flat terrain for the ). This is exacerbated by the fact that AT-AT walkers are using fusion power cells as compared to diesel engines.   It would seem that it would make far more sense for the Empire to remove the legs and mount the main chassis on treads, which would make the vehicle far more effective. Why then, did they not do this? It clearly was not the case that imperial Engineers lacked imagination, as they did produce .   Possible answers which I find implausible as advantages:  1: AT-AT walkers are tall, and therefore have a surveying advantage over the battlefield / have a better vantage point to fire from  Orbital satellites can provide the observation, and in any case air/orbital support should always be called in, especially if fire support from above is required. What's the name of this flat illustration style? I want to buy some stock photos like the ones below. What is this design called?  I don't even know how to start searching... I had Windows 8 installed on my system, which I screwed up, and now I run Ubuntu 12.04. While installing, Ubuntu used all the 461 GB of HDD space (one partition).   Now I want to make a new partition of maybe 100 GB or so, in order to install Windows 7 alongside Ubuntu (with a dual-boot). But I can't even get a partition done. I can't change the partition in GParted either - the option is disabled. Create terrazzo material/pattern with the standard textures is it possible to create this kind of material with the standard textures? The Voronoi texture (set to cells) is coming close but I don’t know if it is possible to a) set the single cells apart from each other and b) to give them seperate colors.