The aim of this paper is to present fixed set theorems, collage type and anticollage type results for single-valued operators T : X × X → X in the framework of a complete metric space X. Based on the coupled fixed point theory, existence of fixed sets, collage type and anticollage type results for iterated function systems are also presented. The results are closely related to self-similar sets theory and the mathematics of fractals. Several examples of coupled fractals illustrate our results.