I need some examples of "prime-like" sets of numbers. Maybe this term is already known by some other standard name. Let me define it. A set $S=\{s_1,s_2,\ldots ,s_n\}\subset \mathbb{R}$, is called "prime-like" if $$\forall \alpha\neq\beta\in \mathbb{N}^n \mbox{ we have} \prod_{i=i}^ns_i^{\alpha_i}\neq \prod_{i=i}^ns_i^{\beta_i}$$
Obviously any set of $n$ distinct prime numbers are "prime-like". One other trivial example is inverse of $n$ distinct prime numbers. Can you provide some other interesting examples of such sets, preferably numbers having magnitudes $\leq 1$. If this definition is known by some standard name in mathematics, please provide the sources.
Moreover, are the following sets $S$ "prime-like"? $$S=\{\cos(1),\cos(2),\ldots ,\cos(n)\}$$ or $$S=\{\sin(1),\sin(2),\ldots ,\sin(n)\}$$