An iterated function system (IFS) is a finite set of contraction mappings on a complete metric space. Symbolically,

$$\{f_i :X \to X \mid i=1,2,..n \},\quad n \in \mathbb{N}$$

is an IFS if each $f_i$ is a contraction on the complete metric space $X$.

I think union, intersection or difference of different IFSs generate new IFSs. This is simply a fact. For example, the union of two IFSs also has an attractor. But, is there any relation between the attractor of union IFS and the attractors of the two IFSs? So, I wonder if it will make sense to use union, intersection or difference of two different IFSs. If it will, how is it interpreted for the attractors or other properties of IFSs?