Definition:Simultaneous Equations/Solution Set

Definition
Consider the system of $m$ simultaneous equations in $n$ variables:


 * $\mathbb S := \forall i \in \left[{1 \,.\,.\, m}\right] : f_i \left({x_1, x_2, \ldots x_n}\right) = \beta_i$

Let $\mathbb X$ be the set of ordered $n$-tuples:
 * $\left\{{\left\langle{x_j}\right\rangle_{j \in \left[{1 \,.\,.\, n}\right]}: \forall i \in \left[{1 \,.\,.\, m}\right]: f_i \left\langle{x_j}\right\rangle = \beta_i}\right\}$

which satisfies each of the equations in $\mathbb S$.

Then $\mathbb X$ is called the solution set of $\mathbb S$.