In mathematics, the term "closed" typically refers to the closure property of a set under certain operations. A set is said to be closed under an operation (like addition, subtraction, multiplication, etc.) if applying that operation to any members of the set results in a member that is still within the set.
Let's break down the definitions you provided:
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Closed Set Meaning: A set is called closed under a certain operation if performing that operation on elements of the set yields results that also belong to the set. For example, the set of integers is closed under addition because the sum of any two integers is also an integer.
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An expression belonging to a closed set cannot be used to create an expression that is not of that set: This is somewhat misleading. It is more accurate to say that if you take two members of the closed set and perform a certain operation on them, the result will still be a member of that set.
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Members of a closed set cannot be added, subtracted, multiplied, or divided: This is incorrect. The key point about closure is that these operations are indeed allowed, but the results must stay within the set if the set is closed under those operations.
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An operation performed on members of that set will result in a member of the same set: This is the correct definition of a closed set under an operation. For example, if you take the set of even numbers, adding two even numbers will always yield another even number, hence this set is closed under addition.
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The set has a finite number of members, as opposed to being infinitely large: This statement is also incorrect in the context of closure. A closed set can be either finite or infinite. For example, the set of all real numbers is infinite, yet it is closed under addition, multiplication, and other operations.
In summary, the accurate definition of a "closed set" pertains to its closure under certain operations, meaning those operations on members of the set yield results that remain within the set.
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