Notation of Relations which are not functions

Question

I am a student in class 11, and have a minor doubt regarding the notation of relations that are not functions. Clarifying this query will surely help me a lot.

I know that a function is represented in the form $f(x)=x^2+2$, where $x$ represents an element part of the domain, and $f(x)$ represents its corresponding image. I was wondering if this notation can only be applied to functions or all relations in general? For example, can I represent a square root relation (not a function since it fails the vertical line test) in the form $g(x)=\pm\sqrt{x^2}$?

Answer 1

No, you should use function notation only for actual functions. Writing "$x=\pm a$" is informal shorthand for "$x=a$ or $x=-a$" (usually in a context where you're solving for $x$ and there are multiple solutions), but an expression like $\pm a$ doesn't make sense as the definition of something. It's much clearer to define your relation as $g=\{(x,y)\mid y^2=x^2\}$.

Answer 2

Let $f$ be a relation on a set S.

The expression $f(x)$ means " the object that is related to $x\in S$ by relation $f$".

The use of a definite description " the so and so" presupposes that the object under discussion (1) exists (2) is unique.

So, such an expression requires relation $f$ to be a function.