# About least upper bound and ordered set

1. Mar 13, 2012

### jwqwerty

I have two questions:

1. Least Upper Bound
ex)
Let A = {x : x∈Q(rational number) , x^2＜2, x＜0} ⊂ S (ordered set),
Then why doesn't the least upper bound (α,which is √2) exist in S?
As long as I know, S is a set which has an order. And it doesn't state whether it belongs to Q(rational number) or R(Real Number)

2. Ordered Set (S)

Whenever I see definition related to least upper bound, greatest lower bound and so forth, I always see 'A⊂S'

For example, let's look at the defintion of 'bounded above' :
Supposed S is an ordered set and A⊂S. If there exists a β∈S such that x≤β for every x∈A, we say that A is bounded above.

So my questions is that why is it important to state A⊂S