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> What does it mean if an operation 'is defined' on a set? Does it mean that any 2 elements combined together using that operation always need to output an element which is also in the same set? For a binary operation f to defined on a set, f(x,y) must exist for every x and y in the set. There is no requirement that f(x,y) itself is in the set. Adding that requirement would mean that the set is "closed" under the operation f. So if we take Z+ = {1, 2, 3, ...}, ordinary division is defined on Z+, but Z+ is not closed under division, since we can get results like 2/3 that are not in Z+. Whereas division is not defined on Z0+ = {0, 1, 2, ...} because we can get undefined results like 2/0. However, some definitions of "binary operation" include the "closed" property, so under such definitions, division would not be considered a binary operation on Z0+. >Does it mean that any 2 elements combined together using that operation always need to output an element which is also in the same set? Specifically in the case of a field, yes; addition needs to be defined, closed and invertible for the set; multiplication needs to be defined, closed and invertible for the set excluding the additive identity (zero). |