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?

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 $R$. 
