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August 26th, 2012, 01:08 PM   #1
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largest set vs smallest set

Hello all

I am new to logics and set theory. When I read some definitions, many refer to ".... X is the smallest set such that... " or "... Y is the largest set such that ..." . What is the difference? Can you refer to tutorials that explain this concept?

Thanking you
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August 26th, 2012, 06:18 PM   #2
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Re: largest set vs smallest set

Well, I'm a novice to Set Theory too, but I guess I can help =D
What I think it may be is that the size of the set is the amount of elements of that set. So if you consider for example all sets that satisfies some given property "The Largest set, namely X, that..." would be the set with the greater number of elements betwen those sets.
In other contexts, let A be a family of sets. There's a theorem that says that there's a ordering in that family (If I'm not missleaded it's called the Well Ordering theorem), like in the natural numbers where 2 > 1 or 3 < 5.

I hope it's right and that you understood xD
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August 27th, 2012, 05:49 AM   #3
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Re: largest set vs smallest set

Note also that it is common to write something like "X is a largest set" where "largest" need not be unique in that situation.
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August 27th, 2012, 04:50 PM   #4
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Re: largest set vs smallest set

A common usage is to use the terms largest and smallest with respect to the partial order of set inclusion. So A is smaller than B if A is a subset of B, and so forth.

For example, if X is a subset of a ring, then the ideal generated by X can be characterized as the intersection of all the ideals that contain X. Since the intersection of a collection of sets is a subset of each of the sets, we can say that the ideal generated by X is the smallest ideal containing X; where "smallest" is defined in terms of set inclusion.
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