I'm given that x, y, and A, B are sets, and must prove that the Cartesian product   A x B := {(x, y) | x $\in$ A and y $\in$ B} is a set.  I'm certain that I need to use AC somehow, but I'm having trouble understanding what exactly it would even mean for me to apply it, or how I should go about it.   Here is what believe to understand about AC: if I have a set, say Z = {a, b, c} then applying AC means that I have a function:  f:Z $\mapsto$ $\cup$Z, i.e., f:{a, b, c} $\mapsto$ {{a}, {b}, {c}}   And we have that $\forall$z $\in$ Z (f(z) $\in$ z)  Unfortunately I'm having trouble seeing the consequences of this and how I can extend it to prove that the Cartesian product between A and B is indeed a set. But it could be as well that I've totally misinterpreted the axiom and so obviously can't draw any meaningful conclusions. I would really appreciate if someone could clear the air for me and hopefully set me in the right direction.
In ZFC set theory, every element object is a set. We know or feel that if $X$ and $Y$ are sets, their cartesian product $X\times Y$ is a set.  My question is: Why if $X$ and $Y$ are sets, their cartesian product $X\times Y$ is a set?
During graduate studies online application submission period, is there any preference given for an applicant if he/she submits his application at the beginning of that period compared with someone who submits his/her application couple of hours before the deadline?  Or as long as the applications are complete there is absolutely no difference! How to make two people play at the same time? My brother owns Skyrim on steam. We use Steam Family Share, but we were wondering how both of us could play it at the same time? combinatorics: sum of product of integer compositions Why can't I associate files with Excel after upgrading to 2010? How to prove $n^5 - n$ is divisible by $30$ without reduction Apply a string replacement on the citation as returned by \cite I have biblatex with options  \usepackage[backend=bibtex, style=authoryear]{biblatex}   The citations made with \cite look like this: Smith, 2019 or Smith and Baker, 2019 and so on. I would like to define a command that automatically replaces the occurrence of my name with my initials. Namely I need  \cite{SmithsPaper}  % --&gt; Smith, 2019 \cite{MyPaper}      % --&gt; Manny, 2019 %%% \newcite{SmithsPaper}  % --&gt; Smith, 2019 \newcite{MyPaper}      % --&gt; MC, 2019   I have tried with the package xstring to do something like  \newcommand{\newcite}[1]{%   \begingroup   \edef\@temp{\cite{#1}}%   \StrSubstitute{\@temp}{Manny}{MC}   \endgroup }   The problem is that I can't figure out how to tell LaTeX to expand \cite{#1}. As a result the citation comes out without the replacement. If I change \cite{#1} by simply #1 and I call \newcite{bla Manny bla} it gives bla MC bla as expected.  It shouldn't be relevant but I'm using this within the beamer documentclass. How to get the entity ID for a custom field in Apex? I am looking for a way to get the 15 character entity ID for a custom field on my Lead object in an Apex controller. 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Thanks How to integrate $\int\sqrt{x^2+1}dx$ $$\int\sqrt{x^2+1}dx$$ I've been attempting this problem for days and I've made very little progress. I understand that I should substitute $x=\tan(x)$ and $dx=\sec^2(x)dx$ but beyond that I am very much lost. why normalize and the definition of directional derivative How to change the forgotten root password? I forgot my root password...actually I think I know what it is because it is the same password I have for pretty much everything but in the Terminal in tell me "Sorry, try again." Is there a way to reset the password? If so how? Solving the recurrence $T(n) = \sqrt n T(\sqrt{n}) + \sqrt{n}$ A former student of mine was TA-ing an algorithms class last quarter and asked students to solve this famous recurrence relation:  $$T(n) = \sqrt n T(\sqrt{n}) + n$$  There are  and this question has already been asked here.  I was talking to my student about this problem and mentioned that it's quite hard to solve and is pretty dependent on the particular choices being made. As an example, I suggested this variant of a recurrence relation that was less simple to solve:  $$T(n) = \sqrt n T(\sqrt{n}) + \sqrt{n}$$  The TA and I tried solving this recurrence for about an hour and a half without making any progress. Here are a few things we tried:   My approach to solving the initial recurrence relation was to draw out a recursion tree and notice that each level of the tree contributes $n$ to the total and that there are $O(\log \log n)$ layers, so the recurrence solves to $O(n \log \log n)$. When I tried doing this here, I noticed that the work per layer was no longer constant; instead, the top layer sums to $\sqrt{n}$, the second layer to $n^{3/4}$, the third to $n^{7/8}$, etc. We got stuck working with the summation $n^{1/2} + n^{3/4} + n^{7/8} + ...$. We tried using the iteration method to unroll the recurrence. With the original recurrence relation, this works out nicely; here, we got stuck at the same summation given above.   I'm completely stuck trying to figure out how to solve this recurrence relation. There's nothing riding on it per se - I don't need to solve it for any particular reason - but the fact that we arrived at it by a straightforward modification of a common algorithms problem set question makes it all the more enticing.  Any idea how to solve this recurrence?  Thanks! Money - Countable or Uncountable noun page suggests that we use much with only uncountable nouns whereas the use of many/several is limited to countable nouns only. So I conclude that money is uncountable noun as I've heard people saying phrases like      So much money!   But, We do count the money in our daily life then why is it that money is considered an uncountable noun instead of countable? What if I say      I have $500 to spend   Wouldn't the money be considered countable in this case?  Edit The question may be considered duplicate of the suggested question but, in my opinion, the question on that link itself is not answered properly. I have a 4GB DDR3 800Mhz recently I installed another 4GB. It turns out it's  DDR3 667MHz. Does it really matter I should own both rams 800MHz?    Slot1 shows recent RAM installation. ALSO I don't know if this is 1600MHz (it shows 800x2 and 667x2 for other) or just 800MHz ram. Repair disk from single user mode (fsck fails) Let  $f(xy) =f(x)f(y)$ for all $x,y\geq 0$. Show that $f(x) = x^p$ for some $p$.    I am not very experienced with proof. If we let $g(x)=\log (f(x))$ then this is the same as  $g(xy) = g(x) + g(y)$   I looked up the hint and it says let $g(x) = \log f(a^x) $   The wikipedia page for functional equations only states the form of the solutions without proof.  Attempt  Using the hint (which was like pulling a rabbit out of the hat)  Restricting the codomain $f:(0,+\infty)\rightarrow (0,+\infty)$ so that we can define the real function $g(x) =  \log f(a^x)$ and we have $$g(x+y) = g(x)+ g(y)$$   i.e $g(x) = xg(1)$ as $g(x)$ is continuous (assuming $f$ is).  Letting $\log_a f(a) = p$ we get $f(a^x) =a^p $. I do not have a rigorous argument but I think I can conclude that $f(x) = x^p$ (please fill any holes or unspecified assumptions) Different solutions are invited Where should looking-for-resources questions go? Questions like looking for css templates, blog templates, icons ... In general, resources (let's leave the licensing terms aside for now) which you might need when building a project of your own. Where should those questions go?  They're definitely not on-topic on Stack', SuperUser maybe although I doubt it. Webapps? Hmm, probably not.  Is there a place where they would be on-topic on the SE network?