Are there any particular term for a pair $(U;A)$ where $U$ is a set and $A\in\mathscr{P}U$? That is, saying informally, $(U;A)$ is a set $A$ together with a set $U$ on which $A$ is defined ($A$ is defined as a subset of $U$).
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For more general situations of this kind, there are indeed names. For example, one can talk about filtration, if you have a set of sets with certain inclusion properties. http://en.wikipedia.org/wiki/Filtration_%28abstract_algebra%29 If the sets are vector spaces, this is a (very trivial) flag. But normally, one does not use this terminology for the simple situation of "a set and a subset". You can view it as an element of the inclusion relation, but there are few contexts where this would be helpful. |
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