Equations

Definition: Equation

Let $X$ and $Y$ be two not necessarily unique sets such that $Y$ has an equivalence relation $E_Y$ defined on it.

An equation over $X$ is an 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 $f(x^{\ast })E_{Y}g(x^{\ast })$.