Notation and major confusion with the definition of the probability simplex So it starts like this:
Given a discrete set $N$, the probability simplex over $N$, denoted $∆(N)$ is defined to be:
$$Δ(N) = \left\{ x \in \mathbb{R}^{|N|} \;:\; x_{i} \geq 0 \text{ for all } i, \text{ and } \sum_{i=1}^{|N|} x_i = 1\right\}$$
I guess I kind of understand what it tries to convey, but in practice something doesn't add up. 
For example, suppose, that we have $x= \{x_1,x_2\}$ with dimentionality $|N| = 3$ according to the specification above.
Say:
that $x_1 = \{\mathbb{R}_1, \mathbb{R}_2, \mathbb{R}_3\}$
and
$x_2 = \{\mathbb{R}_4, \mathbb{R}_5, \mathbb{R}_6\}$. 
How does this summation $\sum_{i=1}^{|N|} x_i = 1$ work? 
 A: Well, what is meant under $\triangle(N)$ is the following: if your vector $x \in \mathbf{R}^{|N|}$ has the following properties: 


*

*$x_i \ge 0$ for all $i$

*$\sum_{i=1}^{|N|} x_i = 1$
Then, your vector is from simplex $\triangle(N)$. The sum is taken over all coordinates of the vector. 
Clearly, if one of these conditions is violated then your vector doesn't belong to simplex $\triangle(N)$.
A: I think, this is a kind of question that should be answered via examples.
For any set $N_2=\{a,b\}$ with 2 elements the following vectors $x_1, x_2, x_3$ belong to $\Delta(N_2)$:
$$
x_1 = 
\begin{bmatrix}
0.2 \\ 0.8
\end{bmatrix}, \quad 
x_2 = 
\begin{bmatrix}
0.4 \\ 0.6
\end{bmatrix}, \quad 
x_3 = 
\begin{bmatrix}
0 \\ 1
\end{bmatrix}.
$$
For each of them we have $x_i \in \Delta(N_2)$. Of course there are many more such vectors.
For any set $N_3=\{a,b,c\}$ with 3 elements the following vectors $x_1, x_2, x_3$ belong to $\Delta(N_3)$:
$$
x_1 = 
\begin{bmatrix}
0.2 \\ 0.3 \\ 0.5
\end{bmatrix}, \quad 
x_2 = 
\begin{bmatrix}
0.4 \\ 0.4 \\ 0.2
\end{bmatrix}, \quad 
x_3 = 
\begin{bmatrix}
0 \\ 1 \\ 0
\end{bmatrix}.
$$
For each of them we have $x_i \in \Delta(N_3)$. Again, there are many more elements of $\Delta(N_3)$.
Note that the set $N$ is actually of no matter for constructing $\Delta(N)$, it is only its size $|N|$ that matters.
