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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.

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# Homework Help: Binary operations, subsets and closure

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