Understanding the Definition of a Directed Set

[ehrenfest]

Homework Statement

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?

Homework Equations

The Attempt at a Solution

 

[Hurkyl]

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?

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.

 

[ehrenfest]

I see. Thanks.

 

[HallsofIvy]

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.-