# Directed sets

1. Sep 12, 2007

### ehrenfest

1. The problem statement, all variables and given/known data
http://en.wikipedia.org/wiki/Directed_set

The definition of a directed set at the site above makes no sense to me. The part that does not make sense is: "for any two elements a and b in A, there exists an element c in A (not necessarily distinct from a,b) with"

If c does not need to be distinct from a or b, why does this add any restrictions on the binary relation because a possible c is always just max(a,b), where max is defined in the natural way?

2. Relevant equations

3. The attempt at a solution

2. Sep 12, 2007

### Hurkyl

Staff Emeritus
We have no guarantee that a and b are comparable (i.e. we may have $a \not\leq b$ and $b \not\leq a$), and thus cannot define a maximum operator.

3. Sep 12, 2007

### ehrenfest

I see. Thanks.

4. Sep 12, 2007

### HallsofIvy

Staff Emeritus
For example consider a colection of SETS with $\le$ defined by $A\e B$ if and only if $A\subset B$. It is quite possible to have A and B, $A\me B$ such that A is not a subset of B and B is not a subset of A.-