This depends on what you mean by "property". Universal properties are preserved "downwards" i.e. in subsets (more properly, substructures). Existential propert

This depends on what you mean by "property". Universal properties are preserved "downwards" i.e. in subsets (more properly, substructures). Existential properties are preserved upwards, i.e. in supersets (extensions). For example, the property "there exists an element that equals itself when added to itself" is preserved in a superset. If you have sentences that mix quantifiers (e.g. "for all epsilon, there is a delta ..." or even just "every element has an inverse"), then all bets are off. The branch of mathematics where this is studied (and rigorously proven) is model theory. edit: I guess by "property" (given the context of the discussion) you might just mean "function" or "method", in which case, yes, a function defined on a set is also defined on a subset but not necessarily vice versa, so there are in a certain sense "more" functions defined on the subset.