I'm not sure after reading definitions and explanations about notation of Dom(f) in books. Which is correct notation for denoting Dom(f) between X, {$x\in X$|(x, y)$\in f$ for some y}, and {$x ∈ X$ : $∃y(y ∈Y$ ∧ (x, y) $∈ f$)}?
3.2 Relations
Let R be a relation from A to B. The domain of the relation R, denoted by Dom(R), is the set of all those $a \in A$ such that $aRb$ for some $b\in B$ such that $aRb$ for some $a \in A$. In symbols,
Dom(R)={$a∈A$| $(a, b)∈R$ for some $b∈B$}
and Im(R)={$b∈B$| $(a, b)∈R$ for some $a∈A$}
(. . .)
3.4 Functions
Definition 8. Let X and Y be sets. A function from X to Y is a triple (f, X, Y), where f is a relation from X to Y satisfying
(a) Dom(f) = X.
(b) If (x, y)$\in f$ and (x, z) $\in f$, then y=z.We shall adhere to the custom of writing f: $X\space \rightarrow Y$ instead of (f, X, Y) and $y=f(x)$ instead of $(x,\space y) \in f$.
Source: Set Theory You-Feng Lin, Shwu-Yeng T.Lin
DEFINITION 4.1.4
Let R ⊆ A B. The domain of R is the set
dom(R) = {$x ∈ A$ : ∃y($y ∈B$ ∧ (x, y) $∈ R$)},
and the range of R is the set
ran(R) = {$y∈ B$: ∃x($x∈A∧(x,\space y)∈R$)}.
Source: A First Course in Mathematical Logic and Set Theory by Michael L O'leary
Best Answer
Definitions in mathematics come in many forms. This is a matter of presentation and chosen settings. Furthermore, what is considered a good definition in mathematics is a matter of tradition, usefulness, convenience, and so on. This is perhaps more a matter of adequacy. The bottom point is: It does not matter which variant of a definition you work with as long as the variants are logically equivalent.
In your particular case, you have two variants. When expressed in the same formal language (choose your preferred notation here), they are:
(1) $\left\{ x\in X\mid\exists y,(x,y)\in f\right\} $
(2) $\left\{ x\in X\mid\exists y,y\in Y\wedge (x,y)\in f\right\} $
The second definition requires y to be an element of some set Y, the first does not. The two variants are therefore not logically equivalent. In practice however, definitions are used within a certain context. Definition (2) is likely to be used in a context that has already declared the range of f to be included in set Y, in which case $(x,y)\in f$ implies $y\in Y$.In such a context, it does not matter then which definitions you use.