Please confirm if my following attempt is correct  Let x ∈ (A − B) ∪ (B − A) ⇒ x ∈ (A – B) ∨ x ∈ (B – A)      ⇒ (x ∈ A ∧ x ∉ B) ∨  (x ∈ B ∧ x ∉ A)   ⇒ ((x ∈ A ∧ x ∉ B) ∨ x ∈ B) ∧ ((x ∈ A ∧ x ∉ B) ∨ x ∉ A)  by distributive law  ⇒ ((x ∈ A ∨ x ∈ B) ∧ (x ∉ B ∨ x ∈ B)) ∧                                  ((x ∈ A ∨ x ∉ A) ∧ (x ∉ B ∨ x ∉ A))                      by distributive law  ⇒ ((x ∈ A∪B) ∧ TRUE) ∧ (TRUE ∧ x ∉ (B ∧ A))  (As  P ∨ ¬P = TRUE)   ⇒  x ∈ A∪B) ∧ x ∉ (B ∧ A)      (As P∧TRUE = P)  ⇒ x ∈ (A∪B) − (B∩A)  ⇒ (A − B) ∪ (B − A) ⊆ (A∪B) − (B∩A)   Conversely,  Let y ∈ (A∪B) − (A∩B)   ⇒  (y ∈ A ∨ y ∈ B) ∧  y ∉ (A∩B)   ⇒ (y∈A ∨ y∈B) ∧ (y∉A ∨ y∉B)  ⇒ ((y∈A ∨ y∈B) ∧ y∉A) ∨ ((y∈A ∨ y∈B) ∧ y∉B)  ⇒ ((y∈A ∧ y∉A) ∨ (y∈B ∧ y∉A)) ∨ ((y∈A ∧ y∉B) ∨ (y∈B ∧ y∉B))  ⇒ (FALSE ∨ y∈(B−A)) ∨ (y∈(A−B) ∨ FALSE)   (As P ∧ ¬P = FALSE)   ⇒ y∈(B−A) ∨ y∈(A−B)      (As P ∨ FALSE = P)  ⇒ y∈((B−A) ∪(A−B))   ⇒ (A∪B) − (A∩B) ⊆ (A − B) ∪ (B − A)   … (ii)  Therefore from (i) and (ii) we get;           (A−B)∪(B−A) = (A∪B) − (A∩B)   Proved
Is $A\triangle B := A\cup B-A\cap B $ a suitable definition for the symmetric difference of two sets $A$ and $B?$ Or is $A\triangle B := (A-B) \cup (B-A) $ only allowed?
How long does it take for police 'presence' to clear when they are alerted? I engaged with some enemy NPCs, and accidentally shot a civilian in the head (though not sure it would've mattered if I hit them somewhere else or not) as they ran across my screen. As a result of this, I was alerted by the game that the NCPD (the police) were automatically en route to 'investigate', and that I could avoid them by leaving the area and &quot;lying low for a while&quot;. How long does it take for the police to stop searching for me, and is there any kind of visual information on the HUD that alerts me to when NCPD is or is no longer looking for me? Unable to lock the administration directory (/var/lib/dpkg/), are you root? I get this error in Lubuntu 12.04 when I try to update in terminal:  $ apt-get update E: Could not open lock file /var/lib/apt/lists/lock - open (13: Permission denied) E: Unable to lock directory /var/lib/apt/lists/ E: Could not open lock file /var/lib/dpkg/lock - open (13: Permission denied) E: Unable to lock the administration directory (/var/lib/dpkg/), are you root? I'm using a curve modifier and an array modifier. My object is at one end of the curve but as soon as I choose the curve for the curve modifier it moves to the center. How can I fix the problem so that I can have the array going from one end of the curve to the other? Why two inverters cascaded in multiplexer control signal terminals? Why cascaded inverters are used in the multiplexer when it is possible to realize the circuit using a single inverter. Markdown for ordered lists, starting with zero? I see that you can create an ordered list using Markdown notation like the following.  1. line 1 2. line 2 3. line 3 ... N. line N   The Markdown content supports starting each line with any arbitrary number and reorders the list starting with 1. I'd like to start an ordered list with zero. My use case for zero would be in describing capture groups from regular expressions where group zero always returns the entire string. So I'd like to display something like this:  # Groups 0. Entire string 1. The first capture group you were looking for 2. The second capture group 3. The third group with the string I heart Kittens   Markdown instead reorders the group and forces the ordered list to start at 1:    Trying to force the issue with inline HTML: &lt;ol start="0"&gt; or &lt;ol&gt;&lt;li value="0"&gt;entire string&lt;/li&gt;&lt;/ol&gt; doesn't work either. Any ideas on how I can create ordered list starting with zero?  If it's not possible to do so, then this is a feature-request. Are matrices and second rank tensors the same thing? Tensors are mathematical objects that are needed in physics to define certain quantities. I have a couple of questions regarding them that need to be clarified:   Are matrices and second rank tensors the same thing? If the answer to 1 is yes, then can we think of a 3rd rank tensor as an ordered set of numbers in 3D lattice (just in the same way as we can think of a matrix as an ordered set of numbers in 2D lattice)? Counting number of vertices of polygons and lines in QGIS? In QGIS, is there a way to count the number of vertices of polygons and lines in a given layer, and create a separate field containing these values?   I suspect it's a function somewhere in the field calculator but I can't find it. I made a skin by using Skincraft and I exported it to minecraft.net. I chose my skin and uploaded it. At the top it says,      "Your skin has been changed! Please wait a few seconds for it to   update."   I went to Minecraft, clicked singleplayer and chose a random world at the top. I looked at my character, but it's still default. Multiplayer won't load for some reason. I don't know what I've done wrong! There is a way to use my Galaxy tab s like a wacom table, on linux? I read some articles, and seen some videos about a github project that do a sort of thing, but it's a little bit outdated things.I wanna see if this thing is possible to do, nowadays. Thanks Assurance that ST_Envelope works with geography as expected Using PostGIS, I have a table of data with geographic points defined as last_known_coordinates GEOGRAPHY(POINT,4326). I want to find all of these within a geographic bounding box with given top left and bottom right geographic coordinates, and be sure that the calculations are geographic, not geometric, e.g. that the results account for the curvature of the earth. The query I'm using contains the fragment: last_known_coordinates &amp;&amp; ST_MakeEnvelope(?, ?, ?, ?, 4326)::geography As you can tell I'm a bit new to this, and I know what I want, but not necessarily how to do it correctly. Other answers hint that this is correct, but I'd like to know for sure: Will this point data and SQL fragment take into account the curvature of the Earth in the bounding box? How to set Oracle's Java as the default Java in Ubuntu? What's the limit of the sequence $\lim\limits_{n \to\infty} \frac{n!}{n^n}$? How do I put a backdrop behind the model that I am creating so that I can compare the picture to the model while editing? As figured out in  localized SE sites must not be displayed in the HNQ's. But  actually is.        Has to be fixed I guess? How to add a custom icon or symbol to a flag link? I'm working on a 'sample request' implementation for a commerce site using , , sessions and a little custom code.  I'd like to add an icon or a symbol to let people know that an item they are viewing has been flagged.  I'm not sure exactly how to approach this, I can imagine perhaps adding a conditional field that displays an image or check box that is then changed by  or ajax, or perhaps something more complicated in JavaScript?  It could really be as simple as adding a checkbox that is checked or unchecked as the flag link is clicked or maybe even changing the css/html on click to be identified as marked.   Anyway, grateful for any advice here. During my studying to integration I find this integration.  So I tried to prove but I got stuk.  So I need help in this integration.  $$\displaystyle\int_{-\infty}^{\infty} \frac{x \sin (x)}{1+x^2}  dx = \frac {\pi}{e} $$ Is it possible to use the startup.blend UI layout in older files? Today I bought an ibook G4 from a garage sale. The people who sold it to me didn't know the password for the 2 accounts on it. Apparently it was the guys late father's laptop. Any suggestions on how to get into the laptop and set a new admin account and password? Converse to Chinese Remainder Theorem So as seen on this question , we know that if $(n,m) \neq 1$, then $\mathbb{Z} /mn \mathbb{Z} \ncong \mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/m\mathbb{Z}$, because the right hand side does not have an element of order $nm$.  But take a more general setting. Let $R$ be a commutative ring, and let $A,B$ be ideals in $R$. $A$ and $B$ are said to be comaximal if $A+B=R$. If $A, B$ are comaximal, then we have: $R/AB \cong R/A \times R/B$. In this setup, is the converse true?      If we have $A, B$ ideals in $R$ such that $R/AB \cong R/A \times R/B$, do we always have that $A$ and $B$ are comaximal? Using singular "they" for an animal As I know,  (not only &quot;it&quot;). Also, if a person's gender is unknown, . Is it possible to use &quot;they&quot; when we talk about a specific animal and in what context?