1. The problem statement, all variables and given/known data 1) Let S be a set and p: SxS->S be a binary operation. If T is a subset of S, then T is closed under p if p: TxT->T. As an example let S = integers and T be even Integers, and p be ordinary addition. Under which operations +,-,*,/ is the set Q closed? Under which operations +,-,*,/ is the set of irrationals closed 2. Relevant equations not sure 3. The attempt at a solution for 1) the irrationals can't be closed under any operations. Q is probably closed under all of them except division. But it is the example and definition that gets me. If T is a subset of the Integers, there is no way it will be closed under addition. Just take the Sup of the set and add 1 or the inf of the set and subtract 1. So I guess my question is: is closure peculiar to the subset we have taken, and if so How can we say that an entire set, like the Integers is closed under some operation.