Let $O$ be the infinite? set of all operations such as addition, multiplication, trigonometric functions, integrals etc, over the independent variables $x$ and $y$.
Let $H_0$ be a subset of $O$.
$$H_1 = \{r \circ t \rvert r,t \in H_0\}$$
$$H_i=\{r \circ t \rvert r,t \in \bigcup_{k=0}^{i-1} H_i\}$$
By $H_1$ I mean all possible compositions of operations in $H_0$. For example:
If $H_0=\{+,*\}$ then the expressions $x+x;x+y;x*y$ would be part of $H_1$ then $x*x*y;x*x+x;x+x*x$ would be part of $H_2$ and so on.
Let $C$ be the set of all curves on the $xy$ plane. We can impose further restrictions on $C$ such as all curves must have no intersections, or be smooth, to have or not have branches (such as the graphs of hyperbolas)
Given a set $H_0$ let $E = \bigcup_{i=0}^\infty H_i$. $E_k(x,y) \in E$.
$E_k(x,y)$ is an expression in terms of $x$ and $y$ obtained by any number of compositions of operations from $H_0$
For the implicit equation $E_k(x,y)=0$ let $I_k$ be the curve of the solution to the previous equation.
$I= \bigcup_{E_k \in E} I_k$
My question: For a given $H_0$ and a set of restrictions on $C$ does $C \in I$?
Is the relation $F:E \mapsto C$, $F:=E_k(x,y)=0$, surjective?
Follow-up and more concrete questions and examples:
1.What is the minimum set $H_0$ for which all graphs of the implicit equations $E_k(x,y)=0$ cover all smooth (with intersections and branches) curves in $C$?
2.Are the common operations such as $+,-,*,/$ and functions such as trigonometric, exponential, logarithmic enough to cover all smooth curves in plane? If not do we need more functions such as the transcendental one Gamma and other similar to it? Maybe even limits and integrals?
3. Given an $H_0$ and a squiggly curve (smooth with intersections and no branches) can we prove the existence of an implicit equation $E_k(x,y)=0$ for which the solution graph $I_k$ is exactly our squiggly curve? Even if we can't find the expression $E_k(x,y)$ can we tell if it exists or not for a given $H_0$?
For all $E_k(x,y)=0$ assume maximum domain $D \in \Bbb R$ where $x,y \in D$ for which $E_k(x,y)=0$ is well defined.
In common language and ignoring a lot of constrains, it all comes down to: Can we map all implicit equations to curves in plane and vice-versa?
Edit
As pointed out by a comment I need to define what do I mean by "curve".
I don't have a proper technical definition but I was thinking of a curve as being any continuous non straight line in plane. It can have branches such as the hyperbola and it doesn't have to be locally differentiable everywhere like a polygon.
But it can have further restrictions such as being of a certain $C^i$ degree of smoothness.
For this question any formal definition of a "curve" can do just fine. I'm interested if the question can be answered for any definition.