Subset Ordering in Order Theory

[jack1234]

I have seen the term "subset ordering of sets" at http://en.wikipedia.org/wiki/Order_theory

What I can understand now is it is something related to the ordering of sets.

But I can't understand literally what "subset ordering of sets" means.
What is the subset, what are the sets, and how they relate to each other?

 

[Hurkyl]

$\subseteq$ is a partial order. (on any class of sets)

 

[jack1234]

Thanks, but I still not really understand the whole picture.
Can you please literally explain what is "subset ordering of sets"?
Very thanks=)

 

[0rthodontist]

In other words, a set A is considered less than or equal to a set B if A is a subset of B.

 

[HallsofIvy]

The "subset ordering" is $A \le B$ if and only if $A \subseteq B$. If $A \subseteq B$ and $B \subseteq C$ then $A \subseteq C$- the transitive property which is the only property required of an order relation.

Are you saying that you don't understand what a "subset" is?

 

[jack1234]

Thanks, now I am getting clearer now=)

 

[HallsofIvy]

Notice that "trichotomy" does not hold: there may be sets A and B such that neither $A\subseteq B$ nor $B\subseteq A$ is true.

(Trichotomy says: Given any A, B, one and only one of these must hold:
1) A< B
2) B< A
3) A= B )