# Can we encode with the set of $\{0,1\}$ and its Boolean operations any finite or infinite domain?

Can we encode with the set of $$\{0,1\}$$ and its Boolean operations any infinite domain that is subset of the real numbers $$\mathbb{R}$$ or the whole set of real numbers? For example can we encode the domain of a random variable $$X$$ that is a subset of the real numbers? Suppose that the random variable is normally distributed with mean $$\mu_x\in \mathbb{R}$$ and variance $$\sigma_x^2>0$$?

• Not clear what do you mean by its boolean operation? Are you trying to enumerate the algebraic equations of finite strings? Does it polynomially bounded? If not, your scheme is already not efficient... Dec 18 '21 at 11:59
• and a random variable $Z$ is not a "domain" in any meaningful way, though something like $\{x: P(Z)\leq x\}$ would be. Dec 18 '21 at 14:11
• Both of you check again my question, I made some changes Dec 18 '21 at 17:06
• $\mathbb{R}$ is uncountably infinite (so is any nonempty interval) but $\{0,1\}^*$ is only countably infinite. Dec 18 '21 at 17:14
• In other words we can not do the eoncodin that i asked for with $\{0,1\}*$...thanks Dec 18 '21 at 17:23