Is "{x | (x, y) ∈ A}" the same thing of "dom A"? I am learning set theory using the book "Axioms and Set Theory" from Robert André and it defines dom R as dom R = {x : x∈C ∧ (x, y) ∈ R for some y∈C}, R is a relation on C. But English ain't my first language and I always find confusing the sentence "for some", but I got what dom R means by the examples. Because of this, I wanted to try using an easier solution (easier for me, of course), and I think I can use the expression dom A = {x : (x, y) ∈ A} instead; I think both expressions have different contexts, but A = {(cake, delicious), (sugar, mehh), (cookie, yummy), (cheese, life)}, would both expression give the same result? Or need I to urgently revise my set theory's knowledge?
P.S.: I am not in/on college, so I ain't a mathematician or studying it formally. I used bold on mathematical expressions because with the white background, it hurts my eyes. 
Thanks for reading my question!
 A: Let's talk about the relation $\leq$ on $\mathbb{N}$ for concreteness.
To understand why $\{n\;|\;n\leq m\}$ is not the same as the domain of $\leq$ , it helps to note that $n$ is a bound variable, but $m$ is a free variable. Binding a variable makes the expression in which it's bound no longer depend on particular values of that variable. Free variables, on the other hand, are like pronouns whose referent hasn't been specified; you don't know what a formula containing them is really on about until you know what it's refering to.
Compare the expressions

There is a prime number less than $x$.

in which $x$ is free, and whose truth value is only determined when we pick some particular value for $x$, and

There exists an $x$ such that there is a prime number less than $x$.

in which $x$ is bound and the sentence is simply true (when talking about natural numbers).
The general idea is the same with set abstracts. In a set abstract, a free variable $x$ means we're implicitly dealing with a sort of cloud of sets, one for each value of $x$, instead of a single unique set. In the case of $\leq$ on the natural numbers, $\{n\;|\;n\leq m\}$ is a different set for each $m$: it's the set of all the $n$ that are less than that particular $m$. So for $m=0$, it's $\{0\}$; for $m=1$, it's $\{0,1\}$; etc.
The expression $\mathrm{dom}(\leq)=\{n\;|\;\exists m(n\leq m)\}$, on the other hand, binds $m$, so the specific set this denotes doesn't depend on a choice of anything. It is the set of all those $n$ which are less than or equal to at least one number; which is all of them (i.e. $\{n\;|\;\exists m(n\leq m)\}=\mathbb{N})$.
So the difference between that "for some $x$" and its absence really is pivotal. Otherwise you have no ability to talk about a set that depends on a particular, but unspecified, $x$, and an existentially quantified $x$ (which are definitely both things you want to be able to do. Just try to remember that (in mathematical writing) "$\varphi(x)$ for some $x$" is typically synonymous with "there is at least one $x$ such that $\varphi(x)$" or "there exists an $x$ such that $\varphi(x)$," phrases which do not use the (very odd, from a natural language standpoint) word "some".
