Thursday, May 11, 2006

More order

Since math and order have arisen here and here, it's time that I introduced you to the Well-Ordering Property. This is one of those incredibly simple statements, but its important particularly when you are talking about sets.

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.

4 comments:

Anonymous said...

And I thougth Well ordering meant you stuck to the low fat side of the menu.

Teresa said...

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.

MathCogIdiocy said...

"low fat side of the menu" *snork*

MathCogIdiocy said...

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.