The Well-Ordering Property: Every non-empty set of positive integers has a least element.
That's it. Notice that it says nothing about the numbers being in order (ie., 1, 2, 3, 4). Only that a set with at least one element (non-empty) will have a least or smallest element.
Before you ask, this property addresses only positive integers because the set of negative integers has no least element. Rememeber that the negative integers go -1, -2, -3, and that -1 is greater than -2 which is greater than -3 and so forth. So while the set of positive integers is well-ordered, the set of negative integers is not.
Math
4 comments:
And I thougth Well ordering meant you stuck to the low fat side of the menu.
I don't suppose it could be turned around for negative numbers.
Every non-empty set of negative integers has a largest element.
Yeah my brain is not not not working today.
"low fat side of the menu" *snork*
Teresa -
The statement "every non-empty set of negative integers has a largest element" is certainly a true statement. However within the grander scheme of number theory, it doesn't necessarily add anything, but instead sets up a situation where the concept of order becomes indidually defined for each non-empty set. There's a loss of generality. Not a good thing.
There's also the fact that the well-ordered property is applied in a more general sense where any non-empty set with a least element is considered well-ordered.
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