Equations

Definition: Equation

Let \(X\) and \(Y\) be two not necessarily unique [[Sets]] such that \(Y\) has an [[Relations|equivalence relation]] \(E_Y\) defined on it.

An equation over \(X\) is an [[../../Logic/Formal Languages|expression]] of the form

\[ f(x) = g(x), \]

where \(f,g: X \to Y\) are arbitrary [[Functions]].

Definition: Solution

A solution to the equation is any \(x^\ast \in X\) such that \(f(x^\ast) \,\, E_Y \,\, g(x^\ast)\).