How to read this mathematical expression?

I am very much interested in machine learning. I would like to do research in this subject. But presently the mathematical language used in this subject is hard for me. Here is an expression in wiki page on Support vector machine :

Given some training data $\mathcal{D}$ , a set of n points of the form

• It may help to say which part of the notation confuses you. For example, do you know what the $\in$ sign means, and are you familiar with the set notation $X = \{x\ |\ x\ \text{has a certain property}\}$? – manthanomen Nov 12 '13 at 4:39
• @manthanomen, the part that confuses me is the $_{i=1}^n$, which I've never before seen in this context. – dfeuer Nov 12 '13 at 4:46
• @ I know 'belongs to' notation etc, but not getting this 'i= 1 to n' – dexterdev Nov 12 '13 at 4:50
• It means that $\mathcal D$ consists of the elements $(\mathcal{x}_i, y_i)$ where $i$ is an index ranging from $1$ to $n$. In other words, there are $n$ elements of the form $(\mathcal{x}_i, y_i)$ satisfying the given properties. For example, you could also write any set $\{x_1, x_2, ..., x_n\}$ as $\{x_i\}_{i = 1}^n$. I believe Mohamed has edited his answer to explain this part of the notation. – manthanomen Nov 12 '13 at 4:59
• Exactly manthanomen! and i added a matrix representation that can explain furthur. Thank's for this remark! – Mohamed Nov 12 '13 at 5:06

That means : $\cal D$ has $n$ éléments $z_1,\cdots,z_n$ having the form: $$z_i=(x_{i,1},x_{i,2},\cdots,x_{i,p},1)$$ or $$z_i=(x_{i,1},x_{i,2},\cdots,x_{i,p},-1)$$ where $x_{i,1},x_{i,2},\cdots,x_{i,p}$ are real numbers.
$$A_{\cal D}=\begin{pmatrix} x_{11}& x_{12}&\cdots& x_{1p}& \varepsilon_1 \\ x_{21}& x_{22}&\cdots& x_{2p}& \varepsilon_2 \\ x_{31}& x_{32}&\cdots& x_{3p}& \varepsilon_3 \\ \vdots &\vdots&\cdots&\vdots& \vdots \\ x_{n1}& x_{n2}&\cdots& x_{np}& \varepsilon_n \\ \end{pmatrix}$$ Where $\varepsilon_i=1$ or $-1$ for all $i \in \{1,..,n\}$
We can now see elements of $\cal D$ as lines of the matrix $A_{\cal D}$.