Can the set $f(A)$ still be referred to as the *image of the set $A$ under $f$* if $f$ is a relation but not a function?

Question

The existence of the inverse image of a set $A$ under $f$, written as $f^{-1}(A)$, does not indicate that $f^{-1}$, the inverse of $f$, exists or that $f$ has any inverse function. In the other direction, can the set $f(A)$ still be referred to as the image of the set $A$ under $f$ if $f$ is a relation but not a function?

Answer 1

Any binary relation have a domain and a range, so yes, you can define the range of a binary relation like this, for any chosen set $A$. In mathematical notation, suppose that $R\subset X\times Y$, then for some set $H$ we can set

$$R(H):=\{u\in Y:(\exists s\in H:sRu)\}$$

Then $R(H)$ is the "image" of $H$ in the range of $R$.