Writing domains: $∈$ or $⊆$? Usually when we write domains for functions (e.g. $f(x)=x^2$) in set notation, we would write something like this:
$$D=\{x∈ℝ\}$$
This means that all values of x are part of the set of real numbers. However, would it not be more appropriate to write
$$D=\{x⊆ℝ\}$$
or
$$D=\{x⊂ℝ\}$$
Because the set of $x$ is a subset of the set of real numbers? Why do we write the domain the first way rather than the second or third ways?
 A: When using the $\in$ (written as $\in$ in Latex), you need an element on the left side and a set on the right. There are many ways of saying this, but when you write $x\in\mathbb R$ you mean $x$ is an element of the set. We shorten this to $x$ is in $\mathbb R$. 
When you're using the symbols $\subseteq$ and $\subset$ they need sets on both sides to compare. Namely, if $A \subset B$, then both $A$ and $B$ are sets, and every element in $A$ also exists in $B$. 
$x \in \mathbb R$ is valid. Listing them is okay too: $a,b\in \mathbb R$. 
An interval, however, is a set, so it needs to be 'compared' to $\mathbb R$. $\mathbb R$ is a set, keep in mind:
$(a,b)\subset \mathbb R$ is a valid thing to write. Note however that mathematical notation sucks, and we need to be careful not to interpret $(a,b)$ as an ordered pair, but rather an open interval. We normally use different letters in the alphabet if we meant to interpret $(a,b)$ as a vector in $\mathbb R^2$.
A: When a set $D$ is specified or desribed by use of brace brackets, it means that the members of $D$ are all those and only those things that satisfy the conditions written between the brackets.
So $D=\{x\in \mathbb R\}$ means that for any $x,$ we have $ x\in D\iff x\in \mathbb R$. Of course that means $D=\mathbb R$.
And if we write $D=\{x\subset \mathbb R\}$ that means that $D$ is the set of all subsets of $\mathbb R.$
To say that every member of D is a real number, write $D\subset \mathbb R$, which says that any member of $D$ is a member of $R$. 
A: This is a rather delicate situation. I had to pause while writing to upvote the question, because it is not as clear cut as I'd initially thought. Let me start with a story (this is going to get much more rambling than I'd intended). 
I mostly teach remedial algebra classes, and always get about 20% of the class writing things like $\{x : x = \Bbb R\}$ for domains of functions, and it makes me cringe! I can't fault the students: they are asked to use notation they don't understand and really shouldn't be pressured to use. But I always feel sad when somebody feels like they need to use symbols (that are clearly misunderstood, as is the case with $\{x: x = \Bbb R\}$) when I would be perfectly happy to see "all real numbers." So, I quietly cross out the "$=$" and write "$\in$" above it. I wish the story had a happy ending.

It would not be correct to say that $x \subseteq \Bbb R$.
At least conventionally. You could define a function whose domain is in fact subsets of real numbers (as is the case writing $\{x : x \subseteq \Bbb R\}$, but it wouldn't be the same as the "squaring function" $f(x) = x^2$ that eats a number $x$ and spits out $x^2$.
When you say that a domain is $\{x : x \subseteq \Bbb R\}$ you are claiming that the domain is in fact a bunch of subsets of $\Bbb R$: So, the function should eat sets of numbers and spit out... well, let's see. You couldn't plug $x = 3$ into your squaring function ($3$ isn't a set, no matter what anybody tells you), you'd have to plug the set $x = \{3\}$ in. In this case, I guess you could say $f(\{3\}) = 9$, or $f(\{3\}) = \{9\}$, either one is reasonable enough. 
But if you plugged $x = \{1, 5\}$ in (it's in the domain, after all) what should $f$ give you back? I would assume the set $\{1^2, 5^2\}$. Which would be fine: It's just not quite the usual squaring function! This is mathematics' best and most frustrating quality: Once you say you're going to do a certain thing (like let functions gorge on sets of numbers, instead of single numbers), you have to live with the consequences -- unless you decide you didn't actually want that at all, in which case you start over.
One objection to writing domains in the way you've suggested is that we already have notation that does something like this: We usually define an image function $f^*$ whose domain truly is subsets of the domain of $f$, so that $f^*(A) = \{f(a) : a \in A\}$. But, mathematicians are lazy, and often we abuse notation: We don't call the new function $f^*$, we often keep calling it $f$, even though the domain has been expanded significantly and is it in truth no longer the same function (and deserves a new name).

So, to summarize: The function most people think of, when you write $f(x) = x^2$, really does have domain $\{x : x \in \Bbb R\}$, and not $\{x: x \subseteq \Bbb R\}$. That doesn't mean there isn't a perfectly good function that seems like the usual squaring function but instead takes sets of numbers, or that you couldn't define a squaring function whose domain is subsets of the real numbers, but they are (subtly!) distinct things, and you probably want the one with domain $\{x : x \in \Bbb R\}$.
