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 $U$).
A set together with its subset
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terminology
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9One could consider this pair as the inclusion $A\subset U$... – 2012-05-20
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0How about *extension*? But I don't think there is a standard term for that. The closer I know of is [pointed set](http://en.wikipedia.org/wiki/Pointed_set). – 2012-05-20
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0You could call it a structure with one unary relation, or simply a unary relation $A$ on $U$. – 2012-10-12