Wikipedia:Reference desk/Archives/Mathematics/2010 April 9

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April 9

Question about vacuous truth

Yes the question above me reminded me of a problem I've been having. If we say "For all x in A, there exist y in A such that x=y", would that be considered true? A here is empty. I know that if we have a "for all x in A, P(x)", where P(x) has no quantifiers, that would be considered true. But in here it has a "there exist" over an empty domain, although of course it comes after the "for all". Money is tight (talk) 04:07, 9 April 2010 (UTC)

Yes, the statement is true. -- Meni Rosenfeld (talk) 06:55, 9 April 2010 (UTC)

Let x be an element of A. OMG, there is no element of A, so this is a contradiction! Which means we can say absolutely anything we want to say about x. --COVIZAPIBETEFOKY (talk) 12:13, 9 April 2010 (UTC)

Informally, this property of the empty set becoms more intuitive if you phrase it as "there is no x in A for which P(x) is false". As there is no x in A - full stop - then this is always true. Gandalf61 (talk) 13:29, 9 April 2010 (UTC)

integer solutions

This question is about a statement in a book I am reading. I apologize if it is too trivial. There is an equation $c_{1}(x_{1}-x_{2})+sjdc_{1}=0$. Here j,d are natural numbers and the rest of the terms are integers, (c1>0). The book says that by letting |x2-x1|=jd|s| we have a solution. We need |s| for some other purpose and cannot simply make the solution in terms of s. My question is how is this the solution? Thanks-Shahab (talk) 05:03, 9 April 2010 (UTC)

Was there any additional context for this equation and the importance of its solutions? Since $c_{1}\neq 0$ this is equivalent to $x_{2}-x_{1}=sjd$. There's no need for absolute values to solve the equation. However, any solution also satisfies $|x2-x1|=jd|s|$, which perhaps is simply what the book needed for the rest of the argument. -- Meni Rosenfeld (talk) 07:19, 9 April 2010 (UTC)

x1 and x2 are supposed to belong to the set $\{a,a+d|s|,\cdots a+nd|s|\}$ and j is among {1,...n}. Sorry for missing that part before. The proof that I am reading is of Rado's theorem (Ramsey theory)-Shahab (talk) 09:17, 9 April 2010 (UTC)

Then it looks like |s| is more important for the argument than s itself, and maybe also that the roles of x1 and x2 are symmetrical. Then indeed it may be that $|x2-x1|=jd|s|$ is a more useful characterization of the solution. -- Meni Rosenfeld (talk) 09:49, 9 April 2010 (UTC)

LaTeX enumerate

\begin{enumerate} \item \end{enumerate}

I know I can label the parts whatever I want by doing \item [(b)] or whatever. But, what if I have several things that I want to be numbered in order and I have something to say in the middle. For example, the definition of a group sometimes is stated A group is a set and a binary operation satisfying the following properties: ... then the 3 properties are listed. Then, "G is said to be abelian if it also satisfies: and property 4 is listed. I have to end up doing two \begin{enumerate}'s to do this and on the second I have to manually input [4.]. That's not a big deal but what if it's more complicated like exercises in a textbook with many different sets of instructions to the different problems but all still need to be numbered in order. I don't want to have to manually type in [7.] through [56.] and what if I add a problem later, I'd have to renumber everything. What can I do? Thanks. StatisticsMan (talk) 20:11, 9 April 2010 (UTC)

I believe that you can use \setcounter{enumi}{4} at the beginning of a new enumerate. (Igny (talk) 22:00, 9 April 2010 (UTC))

Also you can look at \usepackage{mdwlist} to write

\begin{enumerate}
\item 1
\item 2
\suspend{enumerate}
text between enumerates
\resume{enumerate}
\item 3
\item 4
\end{enumerate}

(Igny (talk) 22:09, 9 April 2010 (UTC))

Cool, thanks a lot. I will try both of those out. They both seem like a great improvement over my current method. StatisticsMan (talk) 16:25, 10 April 2010 (UTC)

Eigenvector definition

Outside of the standard "a vector operation on a vector" type definition for an Eigenvector, consider an example in which a matrix is a collection of customer ratings for features of cars that they rented. A set of ratings is used to produce an Eigenvector. What does the Eigenvector represent? Is it proper to say that the Eigenvectors of the ratings simply normalize them so they can be compared? -- kainaw™ 21:09, 9 April 2010 (UTC)

I'm not sure eigenvectors would have much significance in that kind of situation. Say we had a square matrix A where the rows are various cars and the columns are various features. If we have a column vector v, we can think of it as a weighting of the various feature scores and then Av is a vector of composite scores for each type of car using that weighting. The property of v being an eigenvector is a relationship between v and Av, but these two are indexed over different sets: v over different features and Av over different cars, so the fact that they're proportional doesn't say anything useful. Eigenvectors are meaningful when the matrix represents an map from a vector space to itself. Rckrone (talk) 02:42, 10 April 2010 (UTC)

Indeed. An eigenvector is a vector which, when transformed, is proportional to itself. A matrix of ratings like that the OP describes isn't a transformation, so the concept doesn't make sense. --Tango (talk) 15:59, 10 April 2010 (UTC)

I would assume it was meaningless as the example you have given does not constrain the matrix to be square, where as having eigenvectors does. —Preceding unsigned comment added by 129.67.119.26 (talk) 15:45, 10 April 2010 (UTC)

Usually in these situations you compute eigenvectors not of the matrix itself, but of its Gramian. If the $(i,j)$ entry of X is the preference of customer i to product j (after centering and possibly scaling), then the eigenvectors of $X^{T}X$ are components from which you can compose customers, and the eigenvectors of $XX^{T}$ are components from which you can compose products. This is basically just PCA. -- Meni Rosenfeld (talk) 16:58, 10 April 2010 (UTC)

Symbol origins

What are the historical origins of the symbols for addition (+), subtraction (-), multiplication (x) and equal to (=)? I can understand the division one.--79.76.236.198 (talk) 22:25, 9 April 2010 (UTC)

Even though it's on a web site about English usage, this page has exactly the answers you want and appears to be well researched. By the way, if you want to read more about the history of mathematical symbols, Florian Cajori's two-volume book on the subject, published in the 1920s, is a good place to look. --Anonymous, 23:54 UTC, April 9, 2010.