The term "closed" in mathematics refers to specific properties of a set with respect to certain operations. To clarify, here is what "closed" typically means in different contexts:
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Closed under an operation: A set is said to be closed under a particular operation (like addition, multiplication, etc.) if performing that operation on any members of the set always results in a member that is also within the same set. For example, the set of even integers is closed under addition because the sum of any two even integers is also an even integer.
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Not related to finiteness: Being closed doesn't imply that a set has a finite number of members. A set can be infinitely large and still be closed under an operation.
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Expressions and closed sets: This part refers more to the rules of manipulating expressions within a set. If you perform an operation on elements of a closed set, the result should still belong to that set for it to be considered closed.
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Operations allowed: Members of a closed set can indeed be added, subtracted, multiplied, or divided, depending on the operation under consideration and the context of what the set is. The crucial point is that performing these operations results in elements that remain within the same set.
In summary, the correct definition of a closed set, particularly in terms of operations, is that "An operation performed on members of that set will result in a member of the same set."
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