Tienen la misma cardinalidad
No está claro si todos los real/complejo constantes son permitidos, o sólo números enteros. Deje $K$ el conjunto de constantes.
Deje $E_i$ e $E_{ni}$ ser los conjuntos de funciones elementales que, respectivamente, y no tienen la primaria integral y deje $E_a = E_i\ \cup\ E_{ni}$.
El límite inferior de $E_i$ e $E_{ni}$
Puesto que para cada constante $c \in K$, hay una función de $f(x) = c$, tenemos:
$$|E_i| \ge |K|$$
También, para cada constante $c \in K$, hay una función de $f(x) = \sin(1/x)+c$, así:
$$|E_{ni}| \ge |K|$$
Upper bound on $E_a$ if $K = \mathbb{N}$
In this case every elementary function can be written using a text formula, for example sin(2*x)^(x/3). If there were more than countable infinite elementary functions, this wouldn't be possible, so:
$$|E_{a}| \le |K|$$
Upper bound on $E_a$ if $K = \mathbb{R}$
Let's say a form of an elementary function is an elementary function with the constants written as $c_i$. So for example $c_1\cdot x^{c_2}$. Just as in the previous section, we can write every form as a text formula, like for example c1*x^c2. This gives an upper bound of $|\mathbb{N}|$ on the number of forms.
Now to go from a form to an elementary function, we have to fill in the constants. Let's say some form has $n$ constants, then there are $|\mathbb{R}|^n$ ways to choose the constants. It is well known that $|\mathbb{R}|^n = |\mathbb{R}|$.
To get an upper bound on $E_a$, we can multiply the number of forms by the number of ways to choose constants. So $|\mathbb{N}| \cdot |\mathbb{R}| = |\mathbb{R}|$ and we get:
$$|E_{a}| \le |K|$$
Upper and lower bound combined
If $K = \mathbb{N}$ or $K = \mathbb{R}$ probamos estas fórmulas para ser verdad:
$$
\begin{align}
|E_i| &\ge |K| \\
|E_{ni}| &\ge |K| \\
|E_a| &\le |K| \\
|E_a| &= |E_i| + |E_{ni}|
\end{align}
$$
No es difícil ver que se sigue que:
$$|E_{a}| = |E_i| = |E_{ni}| = |K|$$
