> **Definition.** Suppose \\(f : X \to Y \\) is a function between sets. For any \\( S \subseteq X \\) define

$$f_{!}(S) = \\{ y \in Y: y = f(x) \textrm{ for all } x \in S \\} .$$

> This is a subset of \\(Y \\). People pronounce \\(f_{!}\\) as "\\(f\\) lower shriek".

I think there's a mistake in your definition of "\\(f\\) lower shriek" - shouldn't it be "the \\(y\\)s such that every \\(x\\) sent to \\(y\\) is in \\(S\\)"?

$$f_{!}(S) = \\{ y \in Y: y = f(x) \textrm{ for all } x \in S \\} .$$

> This is a subset of \\(Y \\). People pronounce \\(f_{!}\\) as "\\(f\\) lower shriek".

I think there's a mistake in your definition of "\\(f\\) lower shriek" - shouldn't it be "the \\(y\\)s such that every \\(x\\) sent to \\(y\\) is in \\(S\\)"?