mathematical definition of a triangle

Question

If mathematically a square area can be defined using the Cartesian product as $\square : \left( \xi, \eta \right) \in \left[-1,1\right] \times \left[-1,1\right]$, how can I define a triangle using the same terminology? The coordinates of the triangle are $(0,0)$, $(1,0)$, and $(0,1)$.

Answer 1

We can use

$$T=\{(x,y)\in\mathbb{R^2}:y\ge 0\land x\ge 0 \land y+x\le 1\}$$

Answer 2

Using tools I consider to be analogous (which may not be what you have in mind), the analogous definition is actually for the triangle with vertices $(0,0)$, $(0,1)$, and $(1,1)$: it is the subspace of $[0,1] \times [0,1]$ consisting of sorted pairs; i.e. $(x,y)$ with $x \leq y$.

(this is more interesting when you consider higher dimensions, since it reduces down to the two-dimensional aspect; e.g. the condition $x \leq y \leq z$ reduces to imposing the two conditions $x \leq y$ and $y \leq z$)