Quasi-sets (q-sets) are collections resulting from applying axioms, very similar to those for ZFU, to a basic domain composed of m -atoms, M -atoms, and aggregates of these. The axioms of Q include equivalents of extensionality, but in a weaker form, termed "weak extensionality axiom"; axioms asserting the existence of the empty set, unordered pair, union set, and power set ; the axiom of separation ; an axiom stating the image of a q-set under a q-function is also a q-set; q-set equivalents of the axioms of infinity, regularity, and choice. Q-set theories based on other set-theoretical frameworks are, of course, possible.