In mathematics, particularly in the context of set theory and algebra, a set is considered to be "closed" under a particular operation if performing that operation on members of the set always results in another member of the same set.
For example:
- A set of numbers is closed under addition if the sum of any two members of the set is also a member of the set.
- Similarly, a set is closed under multiplication if the product of any two members of the set is also a member of the set.
Here’s a more precise breakdown of your statements in light of this definition:
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The set has a finite number of members, as opposed to being infinitely large.
- This statement is not directly related to the concept of being closed. A closed set can have a finite number of members or be infinite.
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Members of a closed set cannot be added, subtracted, multiplied, or divided.
- This is incorrect. Members of a closed set may indeed be added, subtracted, multiplied, or divided; the key point is that the result of these operations must still be within the set.
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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 a bit ambiguous but points towards the right idea. The crux is that operations performed on members of a closed set should yield results that are also within that set.
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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 closure. If a set is closed under a certain operation, then applying that operation to any elements from the set will yield an element that is also in the set.
In summary, the essence of a "closed" set pertains to the outcome of operations performed on its members and ensuring those outcomes remain within the set itself.