Subset Relation: Is the subset relation a partial order?

Question

I read in a Wikipedia entry (subset in german http://de.wikipedia.org/wiki/Teilmenge):

"Every set is a subset of itself"

But for example, if A is a set of all sets, with maximum 5 Elements, than A isn`t a Subset of itself.

so the subset-relation isn`t a partial order and the wikipedia entry is wrong, isnt it?

Answer 1

The set $A$ is a subset of the set $B$ is every element of $A$ is an element of $B$. Since every element of $A$ is an element of $A$ the set $A$ is a subset of $A$. This shows the subset releation if reflexive.

If $A \subseteq B$ and $B \subseteq C$ then every element of $A$ is an element of $B$ and every element of $B$ is an element of $C$. Pick $a \in A$. Since $A \subseteq B$ we have $a \in B$. Since $a \in B$ and $B \subseteq C$ we have $a \in C$. Recalling that $a$ was an arbitrarily chosen element of $A$ we see that $A \subseteq C$. Thus the subset relation is transitive.

I will leave anti-symmetry as an exercise.

Answer 2

Here subset notation $\subseteq$ is the "inclusive or" statement i.e A may be equal to A.