TITLE: Formal definition of equation and unknowns
QUESTION [8 upvotes]: I was just wondering about the formal definition of equation, I mean in terms of logic and the theory of sets. Suppose for example I wanted to define an equation on $\mathbb{R}$, of course it might be anything instead. I know every number is a set, and also I know that equality of sets has been defined previously. So I often hear people say something like "A linear equation is an object of the form $a_{1}x_{1}+a_{2}x_{2}+ \cdots + a_{n}x_{n}=b$".  But how do you justify the part of "$x_{i}$ is an unknown". Because in that case I would also say that $3(2)+4(5)=0 $ is an equation.

REPLY [3 votes]: I would say that an equation is a logical predicate. That is, an open dictum whos truth value is determined by some input e.g.
$$P(x) \ : \ x + 2 = 4$$
where $x$ belongs to some domain. To solve an equation is to find the (possibly infinitely long) truth table for this predicate.