I'm sure that this is true, because A and B have the same cardinality (number of elements in the set). So there has to be the same number of ordered pairs with C in both cases. But I'm not sure how to exactly prove it since we just started with cardinality.
Suppose that $A, B, C$, and $D$ are sets with the cardinalities related as  $|A|=|C|$ and $|B|=|D|$. Prove that the cardinality of $A\times B$ is equal to the cardinality of $D\times C$.  I know that I must prove the bijection of the function $f: A \times B \to C \times D$, but I am not sure how to say this? Can anyone help me?
According to the microtype manual, Kerning, Spacing and Tracking are not enabled by default in pdftex (>= 1.4). What are the reasons for this? This is a problem from Introduction to Representation Theory by Etingof et al. I have no idea how to approach it. My friend wrote a program that uses Gaussian elimination to find a basis, which starts with x, y, [x, y], [y, [x, y]], ... but freezes once it gets to elements with 9 or more occurrences of x, y.  I expect that there is some way to characterize the nonzero elements of this Lie algebra (i.e., elements that are "immune" to being shown to be zero through repeated application of the Jacobi identity) which could be turned into a way to enumerate them. I don't see what it is, though.  It's also totally unclear to me how to show that some set we know spans the Lie algebra (proven using some criterion as described in the previous paragraph) is actually linearly independent, unless we find a faithful representation... but I have no idea how to find a faithful representation for an infinite-dimensional Lie algebra.  Hints would be greatly appreciated.  Edit: After looking at the partial output, we can form the "straightforward" conjecture that:   If $n \geq 4$ is even, then there is one basis element of grade $n$, namely $\mathrm{ad}([x, y])^{(n-4)/2}[y, [y, [y, x]]]$. If $n \geq 5$ is odd, then there are two basis elements of grade $n$, namely $[x, z]$ and $[y, z]$ where $z$ is the single basis element of grade $n-1$.   I also have a different conjecture, which is mutually exclusive with the above: that if we flatten out all nonzero elements into the form ${\rm ad}(e_1) {\rm ad}(e_2) \ldots {\rm ad}(e_{n-1}) e_n$ using the Jacobi identity, then in the sequences $e_1, \ldots, e_{n-1}, e_n$ and $e_1, \ldots, e_n, e_{n-1}$, any span containing two x's must contain at least two y's and any span containing five y's must contain at least two x's, otherwise we can use the Jacobi identity to rearrange the element somehow to show that it vanishes. If this is true, then the long-range behaviour must be:   For $n \geq 8$ there is exactly one basis element of grade $n$, namely $[... [y, [x, [y, [y, [x, y]]]]] ...]$ (the pattern $y x y$ repeats).   I still don't see an approach to proving that none of the elements in this sequence actually vanishes, though.  Edit 2: Here is the hint given by Etingof in an email:     No, there are no published solutions, since these problems are used as   a homework for classes. This is indeed a nontrivial problem.      This is the positive part of the affine Kac-Moody algebra of type   A_2^2, which is the twisted loop algebra for sl(2). So what you need   to do is to relate this Lie algebra to loops into sl(2). I see from   your post that you are on the right track. Filtering questions by multiple tags I know you can select a tag and then click on unanswered to view the unanswered questions for that tag.  But is there a way to view questions only for a certain set of tags > 1 tag?  Mainly I'd like the union of tag1 with tag2 with tag3 with tag4. But also the intersection of tags would be nice as well. Old usernames stick to Community Wiki posts The camera in Far Cry 3 is useful for tagging enemies from a distance but also includes the ability to take pictures.  This does not seem to do anything useful when used on animals or enemies.  Is there a purpose? Why can a 1 element set be a member of another set but not a subset of it? I have came across this in a textbook:     $\{2\}\nsubseteq\{\{2\}\}$ but $\{2\}\in\{\{2\}\}$   I understand that $\{2\}$ is an element (member) of the other set but considering $\{2\}$ is a set itself, why is it not a subset? Word to describe the way cattle and buffalo consume fodder? How do I get Mediatek MT7630E 802.11bgn Wi-Fi Adapter working? It worked with the pre-installed Ubuntu, but after re-installing Ubuntu dual-boot with Windows 8.1, the WiFi is not working.  Is there a driver for this adapter that will work on Ubuntu 12.04? How can I install Sun/Oracle's proprietary Java JDK 6/7/8 or JRE? I want to install Oracle's JRE and to update to the latest version with the Software Updater when they released. Is there a Ubuntu package that is provided by Canonical or Oracle?  Before release Java 7, I followed  to install Java 6.  But it doesn't work for Java 7. There is no package sun-java7-xxx. How can you install Java 7? How to interpret output from Linux 'top' command? Say person A and person B are having a discussion/argument.   Person A tells person B that his views are arrogant, or closed-minded, or some other derogatory adjective. But in fact it is person A's views that are arrogant, closed-minded, etc. Person B's views are usually the exact opposite of what person A is accusing him of.  This sort of situation happens a lot when discussing politics or religion, for example.  What do you call person A, for calling person B something that he himself is?  I guess, for the most part, person A doesn't actually realize that he is the one that the adjective applies to, and not person B. Maybe there is a different name if person A knows the adjective also applies to him. We are familiar with the nifty fact that given the Fibonacci series $F_n = 0, 1, 1, 2, 3, 5, 8,\dots$ then $0.0112358\dots\approx 1/89$. 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Why does Sass change the format of my colors? Exclude lstinputlisting I want to save paper when I print my LaTeX file. That's why I want to hide all my code, inserted wtih lstinputlisting{}  I already know how to exclude tables and images or math but the lstinputlisting{} is a problem.   Hope someone can help me  \usepackage{comment} \excludecomment{figure} \let\endfigure\relax \excludecomment{table} \let\endtable\relax \excludecomment{align*} \let\endalign\relax \excludecomment{lstlisting} \let\endlstlisting\relax   edit: added lstlisting Is there a C++ decompiler? I have a program in which I've lost the C++ source code.  Are there any good C++ decompilers out there?  I've already ran across . $d = \gcd(c,m),\ ac≡bc\pmod{\! m}\Rightarrow a≡b\pmod {\!m/d}\ $ [Congruence Cancellation & Division Rule] How would you show that if $ac≡bc$ $\mod m$ and $\gcd(c,m)=d$, then  $a≡b$ $\mod \frac{m}{d}$?  Any help would be much appreciated!